Formal definition of a limit

1. Oct 21, 2008

GoldPheonix

I'm enrolled in a Real Analysis course (and an abstract algebra course). These courses are my first taste of abstract mathematics, and it honestly doesn't make a bit of sense.

This is the Real Analysis book's definition of a limit:

A sequence, X, in the real numbers is said to converge to some L, or L is said to be the limit of X, if for every epsilon > 0 there exists some a natural number, Kappa, such that for all n greater than or equal to Kappa, the terms, x(n), of X satisfy |x(x) - L| < epsilon.

What does that mean? Is there any way of actually explaining this, because my expects me to implicitly understand this... And there's not a chance in heck that I do.

2. Oct 21, 2008

Tedjn

I think you meant |x(n) - L| < epsilon in your last inequality.

What do we mean when we say that a sequence of real numbers converges to L? Well, what do we want it to mean? At first, we may think that what we mean is that, if we start listing the numbers in the sequence: x(1), x(2), x(3), ..., that we will eventually find x(infinity) = L. But x(infinity) does not exist in a concrete sense, because we can never list out an infinity of numbers. Thinking a little more, we realize that what we really want to say is that, as n gets larger and larger (i.e. as n tends to infinity), we want x(n) to get closer and closer to L. That fits our intuition that in the limit, x(infinity) = L.

What do we mean by "closer and closer?" Does that mean, for example, that each successive term in the sequence must be closer to L than the previous one? That is, do we want it to be necessary that |x(n+1) - L| < |x(n) - L|? It turns out that this is too restrictive. What about the sequence 3, 4, 100000, 3, 3, 3, ...? Our intuition tells us clearly that this sequence converges to 3, but 100000-4 > 4-3, which wouldn't satisfy that definition. You see, what only matters is what the sequence does as n gets closer and closer to infinity; because it's never possible to reach infinity, there is a lot of wiggle room. The sequence can behave erratically for the first million, the first billion, even the first googol terms so long as it "settles down" as n gets even larger.

But now we see why, in the definition of the limit, there is the phrase "there exists some a natural number, Kappa, such that for all n greater than or equal to Kappa, ...". All we need is for the sequence to "settle down" for sufficiently large n. We don't know how large Kappa should be--it could be one million, one billion, even one googol. But it clearly must exist; otherwise erratic jumps can occur forever, and our intuition tells us that the sequence does not get "closer and closer" to L in any fashion.

So how do we put into symbols the idea of "get closer and closer" and "settle down" that I brought up before? We return to the idea that each successive term must get closer to L and tweak it a little by focusing on the question of "how close?" Intuitively, when n = infinity, we want x(infinity) = L, so that |x(infinity) - L| = 0. When n is not infinity but as it gets closer to infinity (that is, as n -> infinity), we want |x(n) - L| to get smaller and smaller, so that x(n) becomes closer and closer to L. Indeed, for each level of closeness, let's call it epsilon, we want |x(n) - L| < epsilon for n big enough. How big? Well, Kappa(epsilon). I write it like this to show that Kappa is a function of epsilon (the level of closeness). Beyond Kappa(epsilon), the sequence can still behave erratically as long as it remains within epsilon of L. By making epsilon the independent variable, we make rigorous our idea of "settle down," and we can allow for arbitrarily long lengths of erratic behavior as long as there is some cut-off point.

Finally, we want convergence to mean that the difference between x(n) and L eventually become closer and closer to 0, so that in the limit, the difference is 0. This is what we do when we say that for all epsilon > 0, we can find a cut-off point Kappa, such that if n > Kappa, the sequence remains within epsilon distance from L. We can choose epsilon, no matter, how small, and find a Kappa (usually larger as epsilon gets smaller) such that the sequence is bounded by |x(n) - L| < epsilon when n > Kappa.

Hence, "A sequence, X, in the real numbers is said to converge to some L, or L is said to be the limit of X, if for every epsilon > 0 there exists some a natural number, Kappa, such that for all n greater than or equal to Kappa, the terms, x(n), of X satisfy |x(x) - L| < epsilon."

3. Oct 21, 2008

shan

Not to hijack the topic but your explanation was so good and I was also having problems with formal definitions; what is the difference between limits and convergence? It seems to me like you can use them interchangeably like this:

The sequence Sn converges to L as n approaches infinity.
The limit of Sn as n approaches infinity is L.

Is that wrong though?

4. Oct 22, 2008

Tedjn

The two are the same. Saying that a sequence Sn converges to L means that the limit of Sn as n approaches infinity is L.

5. Oct 22, 2008

JeffNYC

Be careful thinking too generally regarding limits/convergence of a series: i.e.

Both series, if you take the limit of the harmonic series 1/n and the other 1/n^2 series, they will both be zero. But the harmonic series diverges while the other series converges. Definitely look into it a bit more.

If you think about 1/n as n gets infinitely large, there will always be some discernible area under it's curve (diverges).

The other series, as you get large, there will be nothing left under the curve.

So, looking at your statement:

The sequence Sn converges to L as n approaches infinity. (1/n diverges)
The limit of Sn as n approaches infinity is L. (lim 1/n as n gets large is 0)

Hope that was accurate enough to be useful.

Here is the test for divergence:

lim an as n -> infinity does not = 0, the series diverges. But if the terms do go to zero, don't say it converges before looking at other tests, as seen above.

Jeff

Last edited: Oct 22, 2008
6. Oct 22, 2008

Tedjn

Jeff, I think you're getting your series and sequences confused :)

7. Oct 23, 2008

HallsofIvy

There is one technical difference between "limit" and "converge". If a sequence goes to infinity, many texts will still say it has a "limit" of infinity but will NOT say that it "converges" to infinity- they will instead say it "diverges" to infinity.