Why Use the Triangle Inequality in Limit Proofs?

[O.J.]

Hey,

i was reading through the proof of limit of sum rule in my textbook, and I've ran across somethin i can't understamd. in the proof th textbook uses the triangle inequality:

|(f(x) - L) + (g(x)-M)} < e

<= |(f(x)-L)|+|(g(x)-M|

and then used the latter part in the rest of the proof. question is, isn't it invalid to use the second part as its LARGER THAN th eprevious sometimes which means larger error ? how are we justified in using it? enlighten me pls

 

[radou]

Hey,

i was reading through the proof of limit of sum rule in my textbook, and I've ran across somethin i can't understamd. in the proof th textbook uses the triangle inequality:

|(f(x) - L) + (g(x)-M)} < e

<= |(f(x)-L)|+|(g(x)-M|

and then used the latter part in the rest of the proof. question is, isn't it invalid to use the second part as its LARGER THAN th eprevious sometimes which means larger error ? how are we justified in using it? enlighten me pls

What exactly do you mean and which part of the proof don't you understand? |a+b| <= |a|+|b|, it's as simple as that.

 

[Galileo]

Hey,

i was reading through the proof of limit of sum rule in my textbook, and I've ran across somethin i can't understamd. in the proof th textbook uses the triangle inequality:

|(f(x) - L) + (g(x)-M)} < e

<= |(f(x)-L)|+|(g(x)-M|

and then used the latter part in the rest of the proof. question is, isn't it invalid to use the second part as its LARGER THAN th eprevious sometimes which means larger error ? how are we justified in using it? enlighten me pls

You have to show that given an \epsilon&gt;0 you can make |(f(x) - L) + (g(x)-M)| smaller than \epsilon. The premise is that you can make |f(x)-L| and |g(x)-M| as small as possible, so you can choose to make each of the terms smaller than the given epsilon/2.

 

[arildno]

Remember that IF |f(x)-L|+|g(x)-M)|<epsi, THEN we also have that:
|f(x)+g(x)-(L+M)|<epsi, due to the triangle inequality.

 

[mathwonk]

the thing that bothered me about these "proofs" is the logical nonsense involved in assuming real numbers exist with the usua lproperties of numbers, before treating limits.

i.e. by definition a real number is an infinite decimal, and you can't evena dd infinite decimals without limits. Or if you like, equivalently a real number is the limit of a sequence of rationals.

so to even add the real number, you add the entries in the two sequences of rationals and then take the limit of the sequence of sums.

so the theorem that the limit of sum is the sum of the limits is needed to even check that this definition of addition agrees with the old one on rational limits.so this theorem precedes the introduction of real numbers, not the other way around. so the whole exercise in the textbook is logically fraudulent nonsense.

why do people keep writing textbooks that make no sense just because that is the way it was done in the previous generatyion of textbooks?

i mean how is a poor student going to understand anything when the presentation pays no heed to the logic of the subject? if we throw common sense and logic to the winds in explaining the stuff, how is it possible for anyone to amkke sense of it?I mean, are we just hoping no student ever asks "Excuse me Mr Emperor, but if real numbers are so complicated, how do we even add real numbers, or if we can, how do we know the distributive law holds? I mean I can't start adding those infinite decimals at the right end because there is no right end!

Should'nt we understand what a real number is, before making outlandish claims about them like the existence of maxima and minima for real functions?"

And if you say well we are just approximating, so think of a real number as just whatever your calculator gives you, then there is a problem that none of the theorems in the course are true any longer!

Once we acknowledge that a real number is some kind of idealization of what the calculator gives us, we need some theorems abut how these idealizations behave.

since those same theorems are what we then try to impose later, it seems more honest to me to use the introduction of real numbers as an occasion to motivate the need for them. no need to prove them, but at least point out their role in maing sense of everything.

the presentation in most books is backward, there is a pretense of rigor, when the gap in the logic is big enough to drive a truck through.

 

[mathwonk]

and another thing:

the absence from most books even of an explanation of the basic equivalence relation on infinite decimals, i.e. how to add real numbers as decimals, leads to endless and depressing controversies here such as why .9999... = 1.

congratulations to you by the way OJ for trying to understand this.

 

[Crosson]

i.e. by definition a real number is an infinite decimal, and you can't evena dd infinite decimals without limits. Or if you like, equivalently a real number is the limit of a sequence of rationals.

Alternatively, a real number is defined as an element of the unique complete totally ordered field, from which any two elements can be added by closure.

 

[mathwonk]

aren't you assuming existence of such a field? if they do not exist, most calc students are not interested in them.

this is the logical gap i am discussing, math books assume a non trivial existence theorem which requires limit theory and them proceed to develop limit theory afterwards.

 

[arildno]

Well, is that really much different from assuming the existence of some game with a specified set of rules, and on basis on that deduce what valid games could be played?

I don't see that the actual construction work of a particular field has any more relevance than that through the construction one will see that issues of, say, logical consistency in the constructed field are essentially reducible to the issues of logical consistency within the set of more primitive axioms (i.e, from which the construction can be made).

And that, of course, is an important insight, to be sure...

 

[mathwonk]

these comments suggest to me that you guys have not tried for 40 years to teach calculus to an average class of non math majors. It didn't bother me either when I was a young math student.

But I am afraid that our faith in this sort of abstract nonsense really does disqualify those of us who are willing to accept these axioms without evidence they are consistent, from being teachers.

 

[arildno]

these comments suggest to me that you guys have not tried for 40 years to teach calculus to an average class of non math majors. It didn't bother me either when I was a young math student.

But I am afraid that our faith in this sort of abstract nonsense really does disqualify those of us who are willing to accept these axioms without evidence they are consistent, from being teachers.

Interesting!

So, your experience is unequivocally that we SHOULD, indeed, "belabour" the students with the formal construction of the various number systems in order to develop their understanding of the subject?

 

[mathwonk]

you seem to have ignored reading my remarks, where i said there is no need to prove them, but it seems prudent to call attention to the need for proof.

I see I have stirred a hornets here by pointing out the absurdity of the tenets we mathematicians have all bought into so vigorously.

a real human being, as opposed to a brainwashed mathematics student, cares not only whether a limit "exists" but how to find a good approximation to it. In that situation theorems like "every bounded increasing sequence has a limit" are of no use.

One can only approximate limits for sequences which actually get predictably closer to the limit, or as Errett Bishop was fond of saying, "stupid sequences".In my experience, most students are lost already when one introduces the letter"epsilon" simply because it is Greek. I.e. it satisifes the sense of "well that's greek to me!".If in fact the average student cannot even imagine that a letter like "epsilon" represents a small number like .0000...1, then they certainly do not have any idea of what a complete archimedean ordered field entails, (I assume you meant lub complete ordered field, as opposed to cauchy complete, which as you know is not unique.)

I am speaking from experience of teaching calculus almost continuously since 1968.

 

[arildno]

I agree that we shouldn't sweep under the rug those issues that actually should be proven, but where the actual proofs lie a bit too far away from the actual concerns of the particular subject lectured on. At the very least, the interested student should be pointed to where such proofs are conducted.

 

[arildno]

But, mathwonk:
Wouldn't you say that it is crucial that the student is able to dissociate the two ideas:
a) "Does a limit exist?"
b) "How do we "find" a limit, given that it exists?" (say, in terms of some particular algorithm that works in a few special cases)

Too many students are simply not willing to stop for a minute to ponder the a)-question, and will mix this together with b)-issues..

 

[mathwonk]

well i really don't know how to teach it myself either. my experience really seems like 40 years of unsuccessfully trying everything i could think of.but in this day and age when my students often seem to think the laws in the book, like (af+bg)' = af' + bg', only apply to integers a and b, it dawned on me that they do not know what real numbers are at all.

and since they all carry around calculators often with at most 12 place accuracy, it seemed reasonable to begin by trying to explain to them the relation between the numbers on their calculators and the real numbers occurring in the statements of the theorems in the book.

so i try to say that real numbers are some sort of idealization of calcualtor numbers where the number of places is as large as you like.

I do not want to suggest I have anythign at all in common with grothendieck as a mathematician, but i do identify with some of his experiences as ateacher. reading his "esquisse d'un programme" one encounters soemthing like the following:"Upon trying to teach university students, who often come equipped with rather modest mathematical baggage, or even less than modest, I found myself obliged to begin my explanations with objects so simple that they are familiar even to children from their play..."

 

[mathwonk]

yes i do agree that existence and computation should be separated if possible. this is hard to do in practice, as without a notion of what real numbers are, and why they are so hard to pin down exactly, an average student simply cannot conceive of how the two statements could be dissociated.

 

[arildno]

Well, I naively thought that the "number line", extensively expounded upon, would be an adequate visualization of the real numbers in order to start working with these numbers. (For example in how a tenth part can be subdivided in hundredth parts and so on, ad infinitum).
(Of course, I realize that the distinction between rationals and irrationals isn't particularly obvious from such a treatment)

perhaps I'm wrong in this..

 

[mathwonk]

noi like that too. that's my favorite way to explain real numbers,a s it shows why they correspond to decimals and also why .999... = 1.0000but it isn't easy to get across. when i draw a single unit length, say 18 inches long on the board, then two or three subdivisons into tenths pretty much makes a chalk continuum as far as the eye can see, and then we are back to imagination power.

i tried this friday in fact. there is a big leap from 2 decimal place accuracy to infinite accuracy.

and again, if they accept infinite decimals, how do you add them? you can't start at the right. how do you explain that?

if we think this should be easy for students, have you ever had a student say that sqrt(2) actually equals 1.414? I have. they were blown away when i squared it out and didnt get 2.

and when i pointed out that no finite decimal could ever square out to give 2, one asked how i knew that!

maybe one way to motivate this would be to use pythagoras to argue that the diagonal of a square has length sqrt(2), and then use your subdivison approach to argue that then that length shoul have some decimal expansion, which however must be infinite.
then make the jump to saying real numbers are infinite decimals.here is an experience i had myself that shows how tricky it is to assume th axiomatic approach suffices. our prof handed out axioms for the reals and then asked us which ones held or if any failed for the artionals. i said none failed because i could not see any that did.

of course the completeness axiom failed, and he pointed out that if none did fail, then the reals would equal the rationals. i had not appreciated this obvious fact, and hence even a pure math guy like me was essentially saying that under the axiomatic approach i could not see the difference betwen a real and a rational!

fortunately our prof saw the usefulness of asking this question as an exercise.

 

[arildno]

Well, I would have liked that students should understand that saying 5+7=12 means that they can substitute (5+7) whenever they encounter the number "12".

That is, our denary/decimal representation of real numbers is by no means the ONLY valid representation of them, but arguably the "nicest" one, in the sense that we by such a representation might easily determine which of two numbers is the largest one.

However, I appreciate that it is, indeed, a mental hurdle, perhaps THE mental hurdle, to get to the level of understanding that the "same" number may have different, equally valid, representations.

As for addition of infinite decimals, I would suggest an algorithm based on adding FINITE truncations/rounded numbers, along with an explicit error term. The students should then be encouraged to understand that in this way, we can see that the error term gets smaller and smaller, the more decimals we choose to retain.

 

[arildno]

noi like that too. that's my favorite way to explain real numbers,a s it shows why they correspond to decimals and also why .999... = 1.0000


but it isn't easy to get across. when i draw a single unit length, say 18 inches long on the board, then two or three subdivisons into tenths pretty much makes a chalk continuum as far as the eye can see, and then we are back to imagination power.

Yeah, I know..

maybe one way to motivate this would be to use pythagoras to argue that the diagonal of a square has length sqrt(2), and then use your subdivison approach to argue that then that length shoul have some decimal expansion, which however must be infinite.
then make the jump to saying real numbers are infinite decimals.

Well, it has to be coupled with Euclid's proof of the irrationality of sqrt(2), I think. (But that was probably implied by you)

here is an experience i had myself that shows how tricky it is to assume th axiomatic approach suffices. our prof said that he ahd ahnded out axioms for the reals and then asked us which oens held or if any faield for the rtionals. i said none failed because i could not se any that did.

of the completenes axiom failed, and he pointed out that if none didthe the reals would equal the rationals. i hadnot aprpeciated this obvious fact, and hence even a pure math guy like me was essentilly saying that under the axiomatic approach i could not see the difference betwen a real and a rational!

fortunately saw the usefulness of asking this exercise.

Yes, the distinction between the reals and the rationals is indeed a subtle one; I remember I thought it was "fun" that the rationals constituted a line where you couldn't see any holes in it, even though that it was even more holey than a Jarlsberg cheese.
I guess other students got frightened by that idea..

 

[mathwonk]

"As for addition of infinite decimals, I would suggest an algorithm based on adding FINITE truncations/rounded numbers, along with an explicit error term. The students should then be encouraged to understand that in this way, we can see that the error term gets smaller and smaller, the more decimals we choose to retain."

Yes i agree, and that is what I did last week. I said ok if we have an approximation for pi which is good to within 1/10, what is the error in using twice that approximation as an approximation to 2pi?

then we did it for the question of how close do we have to get to pi so that squaring our approximation is within 1/10 of (pi)^2?

It was quite difficult for the class to even hazard a guess on any of these matters as probably no one had ever asked them to think about what it means in practice to say that the sum of the limits is the limit of the sum, and the product of the limits is the limit of the product.

I.e. if you believe that, presumably you will want to know how close x must be to A, before x^2 is within 1/10 of A^2.

It was rather surprizing to everyone that squaring (1000.1), or even 1000.01, does not give a very good approximation to (1,000)^2.and yet some grasp of this sort of thing is needed to really understand the limit laws.

our book just lists them without proof, and asks the student to memorize them. that seems futile to me.

this might seem odd by the way, but in practice it is very hard for many students to realize that 1/n approaches zero as n approaches infinity.

I used to think it was because they did not understand fractions, i.e. that the reciprocal of a big number is small, but this time i hypothesized it is because they do not realize a sequence does have to actually reach its limit.

I.e. it seemed possible that since they know 1/n never equals zero, they deduce the limit cannot be zero.

So I emphasized that the sequence is made up of approximations, and usually none of them is ever perfect. then that linked back to my idea of saying a real number is essentially just a sequence of rationals.

I may be on the wrong track here, but it seemed maybe more understandable to habitual calculator users, to suggest that real numbers do not actually exist in the world, but are best thought of as perfect idealizations represented imperfectly by sequences of rational approximations.

maybe it would be further improved by taking a more thorough algorithmic approach as you suggest.

we will see how this goes. so far so good, but it is early yet.

 

[RocketSurgery]

the thing that bothered me about these "proofs" is the logical nonsense involved in assuming real numbers exist with the usua lproperties of numbers, before treating limits.

i.e. by definition a real number is an infinite decimal, and you can't evena dd infinite decimals without limits. Or if you like, equivalently a real number is the limit of a sequence of rationals.

so to even add the real number, you add the entries in the two sequences of rationals and then take the limit of the sequence of sums.

so the theorem that the limit of sum is the sum of the limits is needed to even check that this definition of addition agrees with the old one on rational limits.


so this theorem precedes the introduction of real numbers, not the other way around. so the whole exercise in the textbook is logically fraudulent nonsense.

why do people keep writing textbooks that make no sense just because that is the way it was done in the previous generatyion of textbooks?

i mean how is a poor student going to understand anything when the presentation pays no heed to the logic of the subject? if we throw common sense and logic to the winds in explaining the stuff, how is it possible for anyone to amkke sense of it?


I mean, are we just hoping no student ever asks "Excuse me Mr Emperor, but if real numbers are so complicated, how do we even add real numbers, or if we can, how do we know the distributive law holds? I mean I can't start adding those infinite decimals at the right end because there is no right end!

Should'nt we understand what a real number is, before making outlandish claims about them like the existence of maxima and minima for real functions?"

And if you say well we are just approximating, so think of a real number as just whatever your calculator gives you, then there is a problem that none of the theorems in the course are true any longer!

Once we acknowledge that a real number is some kind of idealization of what the calculator gives us, we need some theorems abut how these idealizations behave.

since those same theorems are what we then try to impose later, it seems more honest to me to use the introduction of real numbers as an occasion to motivate the need for them. no need to prove them, but at least point out their role in maing sense of everything.

the presentation in most books is backward, there is a pretense of rigor, when the gap in the logic is big enough to drive a truck through.

So is this chapter in Apostol (the proof the OP stated is section 2.29) useless? It preceeds a discussion and proof of Bolzano's theorem so perhaps it makes sense in that context.

 

[HallsofIvy]

I can't help but wonder what O.J., the original poster, makes of all this discussion!

 

[RocketSurgery]

I've been thinking a lot about what Mathwonk said and now I'm confused even more. Now I don't know whether the Reals or the limit concept is more fundamental because I just realized I'd need to prove each using the other (at least at this stage).

 

[HallsofIvy]

Depends on your point of view. You can, of course, talk about limits using only the rational numbers but then you are much more limited: Cauchy sequences do not necessarily converge, bounded monotone sequences do not necessarily converge, etc.
For example, the sequence of rational numbers 3, 3.1, 3.14, 3.1415, 3.14159, 3.141592, etc. is both an increasing sequence having upper bounds and a Cauchy sequence but does not converge to any rational number.

 

[mrandersdk]

I've been thinking a lot about what Mathwonk said and now I'm confused even more. Now I don't know whether the Reals or the limit concept is more fundamental because I just realized I'd need to prove each using the other (at least at this stage).

I would say that the limit concept is more fundamental, because as HallsodIvy say, you can define limits in the rationals. Then you can define a new space as the space of sequences (M) in the rationals, and then make some equivalence relation on this new space (~), and then define the reals as M/~. If you are interested in this construction of the reals, i have a danish note i can translate.

 

[HallsofIvy]

Actually, there are several ways of doing that. One is to start with the set of all non-decreasing, bounded, sequences of rational numbers, say that {a_n} is equivalent to {b_n} if and only if the sequence {a_n- b_n} converges to 0, then define the real numbers to be the equivalence classes for that equivalence relation. That make the "monotone sequence" property, that any non-decreasing sequence having an upper bound converges, easy to prove.

Or you can do the same thing with the set of all Cauchy sequences. Doing that makes it easy to prove Cauchy's criterion.

Of course, in either case, we identify the rational number r with the equivalence class containing the sequence a_n= r for all n.

But I'm not sure I would say that "the limit concept is more fundamental" because it is possible to define the real numbers as Dedekind cuts of rational numbers, not using limits, explicitely, at all.

 

[mrandersdk]

you are right, I meant more fundamental in the way, that limits makes sense in absence of the real numbers. But you are right that the real numbers also makes sense in the absence of limits, so i guess it is not more fundamental.

 

[Pathway]

I may be on the wrong track here, but it seemed maybe more understandable to habitual calculator users, to suggest that real numbers do not actually exist in the world, but are best thought of as perfect idealizations represented imperfectly by sequences of rational approximations.

That's incorrect, though. They appear naturally in a geometric context. Suppose you have a black hole of a given circumference C. How far away do you need to stay from its center?

It seems clear to me that that question should have a precise answer (C/2pi), even if measurement error means you can never be assured that your answer is correct to infinite precision.