Proof that two equivalent sequences are both Cauchy sequences

In summary, the sequence a_n-b_n is equivalent to the sequence b_n-a_n iff for each ##\epsilon>0##, there exists an N>0 such that for all n>N, ##|a_n-b_n|<\epsilon##.
  • #1
yucheng
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Homework Statement
Given ##(a_n)_{n=1}^\infty## and ##(b_n)_{n=1}^\infty## are equivalent, proof that ##(a_n)_{n=1}^\infty## is a Cauchy sequence iff ##(b_n)_{n=1}^\infty## is a Cauchy sequence. (Tao's Analysis 1, Exercise 5.2.1)
Relevant Equations
N/A
Let us just lay down some definitions. Both sequences are equivalent iff for each ##\epsilon>0## , there exists an N>0 such that for all n>N, ##|a_n-b_n|<\epsilon##.

A sequence is a Cauchy sequence iff ##\forall\epsilon>0:(\exists N>0: (\forall j,k>N:|a_j-a_k|>\epsilon))##.

We proceeded by contradiction. Suppose ##(a_n)_{n=1}^\infty## is not a Cauchy sequence, this means that there exists ##\epsilon## such that at least one pair ##|a_j-a_k|>\epsilon##, ##\forall N>0 \land j,k>N##.

Thus, let us choose an ##\epsilon## that fulfils the above statement.

Since both sequences are equivalent, ##|a_n-b_n|<\epsilon##.
Since ##(b_n)_{n=1}^\infty## is a Cauchy sequence, we can find a N by definition that fulfils the ##\epsilon## we have chosen.

With this N, we choose a ##|a_n-a_k|>\epsilon##, where n,k > N, from sequence ##(a_n)_{n=1}^\infty##.

From ##(b_n)_{n=1}^\infty## , we also choose ##|b_n-b_k|<\epsilon##, n,k > N.

We arrive at ##|b_n-b_k|<|a_n-a_k|##. However, I am now officially stuck. Thanks in advance.
 
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  • #2
There's no need to argue by contradiction.

Intuitively, you know that ##|a_n-b_n|## is small for large ##n## and also ##|a_p-a_q|## is small for large ##p,q##. You want to show that ##|b_p-b_q|## is small for large ##p,q##. Do you see why you can use the triangle inequality here? Hint: ##b_p-b_q=(b_p-a_p)+(a_p-a_q)+(a_q-b_q).##
 
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  • #3
Infrared said:
Do you see why you can use the triangle inequality here? Hint: ##b_p-b_q=(b_p-a_p)+(a_p-a_q)+(a_q-b_q).##

Is there a way to prove it without resorting to the triangle inequality?
 
  • #4
I don't think so, but why would you want to avoid the triangle inequality? You're not going to get far in an analysis class without it.
 
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  • #5
Infrared said:
I don't think so, but why would you want to avoid the triangle inequality? You're not going to get far in an analysis class without it.
Oh really? I am not even acquainted with it! Thanks for your advice. I'll make sure I learn it. ;-)
 
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  • #6
Infrared said:
Do you see why you can use the triangle inequality here? Hint: ##b_p-b_q=(b_p-a_p)+(a_p-a_q)+(a_q-b_q).##

(Me after staring at the screen for some time) Let me try. Given ##(a_n)_{n=0}^{\infty}## and ##(b_n)_{n=0}^{\infty}## are equivalent, this implies that for some N>=0, there exists ##|a_n-b_n|\leq \epsilon /2## (Let's brush over this little assumption). Now, we see that ##b_p-b_q=(b_p-a_p)+(a_p-a_q)+(a_q-b_q)\leq |b_p-a_p|+|a_p-a_q|+|a_q-b_q| \leq \epsilon /2+\delta+\epsilon /2 = \epsilon +\delta## by the triangle inequality, thus we can rewrite the RHS as ##RHS \equiv \epsilon##, by the definition of ##\epsilon##, we conclude that ##b_p-b_q \leq \epsilon##.

I hope this is complete :)
 
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  • #7
Note quite. You haven't said what your ##\delta## is or how it relates to ##\varepsilon##; it looks like you are saying ##\varepsilon+\delta=\varepsilon##? What you want to prove is just: for any ##\varepsilon>0##, there exists ##N\in\mathbb{N}## such that ##|b_q-b_p|<\varepsilon## when ##p,q>N##. It seems to me that the simplest way to do this would be to make each of the three terms in the above sum smaller than ##\varepsilon/3.##
 
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  • #8
The main idea is that you have two sequences. They are equivalent. We want to show that if one of them is Cauchy, so is the other. What does it mean intuitively for a sequence to be Cauchy? It means that somewhere after a large enough N (a term in the sequence) ANY TWO points of the sequence will be very very close to each other ( if we want the points to be closer, we can just move the N further along).

So if we first assume (an) is Cauchy, WTS (bn) is Cauchy. Use Infrared suggestion in post 7.

Look at your post #6. What do you know about |ap- aq|? Hint. It is your assumption.

What do you know about |bp-ap|? it is also your assumption. What else is missing? I think a better question to ask you. Do you understand what the definition of Cauchy is?
 
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  • #9
Intuitively is easy. If they are equivalent it means that the terms of one are very close to the respective terms of the other. So if one is Cauchy and we have ##|a_n-a_m|<\varepsilon## then we will also have ##|b_n-b_m|<\varepsilon## because ##b_n## is close to ##a_n## and ##b_m## is close to ##a_m## so kinda of $$|b_n-b_m|\approx |a_n-a_m|<\varepsilon$$.
But of course the neatest way to prove the above is by triangular inequality as post#2 suggests very elegantly.
 
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Related to Proof that two equivalent sequences are both Cauchy sequences

1. What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers where the terms get closer and closer together as the sequence progresses. This means that for any small distance, there exists a point in the sequence where all subsequent terms are within that distance from each other.

2. What does it mean for two sequences to be equivalent?

Two sequences are considered equivalent if they have the same terms in the same order. This means that they will have the same limit as the number of terms approaches infinity.

3. How can you prove that two equivalent sequences are both Cauchy sequences?

To prove that two equivalent sequences are both Cauchy sequences, you can use the definition of a Cauchy sequence. This means showing that for any small distance, there exists a point in the sequence where all subsequent terms are within that distance from each other. Since the two sequences are equivalent, they will have the same terms in the same order, making it easier to show that this condition holds true for both sequences.

4. What is the importance of proving that two equivalent sequences are both Cauchy sequences?

Proving that two equivalent sequences are both Cauchy sequences is important because it shows that the two sequences have the same limit as the number of terms approaches infinity. This is a fundamental concept in analysis and is used to prove many important theorems in mathematics.

5. Are all equivalent sequences also Cauchy sequences?

No, not all equivalent sequences are also Cauchy sequences. While two equivalent sequences will have the same limit, they may not necessarily satisfy the definition of a Cauchy sequence. For a sequence to be Cauchy, it must have the property that for any small distance, there exists a point in the sequence where all subsequent terms are within that distance from each other. This condition may not hold true for all equivalent sequences.

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