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Prove that a sequence converges by passing to a subsequence

Quick description

A common task in mathematics is to prove that a given sequence f_n converges. One idea that sometimes helps is to pass to a subsequence of f_n that has better properties than the original sequence. This article discusses various implementations of this idea.

General discussion

  • Suppose we are able to prove that \{f_n\} is a sequence in a totally bounded set in a complete metric space. Then it is natural to look at the subsequences of f_n because we know that any subsequence f_{n_k} of f_n has a convergent subsubsequence. If for every subsequence we can find a subsubsequence converging to the same function f then we know that f_n converges to f as well. This method is explained in the article To prove that a sequence converges, find one or more convergent subsequences.

  • Another way to use subsequences to prove convergence is to pick a particular subsequence whose rapidity of convergence is relatively easy to derive and understand and which we expect to approximate the whole sequence well. Once it is shown that this subsequence converges one proves that the rest of the sequence follows this subsequence closely. This method is explained in the article To prove convergence, find a rapidly converging subsequence.


Convergence title

I like "Prove convergence using subsequences" (just one 'sub') for the title. It encapsulates well the general idea. Another: "There is more than one way to prove convergence using subsequences".

It's not clear to me whether

It's not clear to me whether this should be a single article with various different ways of using subsequences, or whether each different way deserves its own article. This is a general question about the whole of the Tricki – at what point does one split an article up? Another question: how should the first technique relate to How to use the Bolzano-Weierstrass theorem?

subsequences and convergence

The two types of arguments that this article lists are very common in probability theory. They are also of different nature and they usually appear in different contexts.

The first one, the one that goes to the subsubsequences is a soft analysis method, I have seen it used to prove weak convergence in function spaces very often (for example see the large deviations book by Dupuis and Ellis). In this setup, you have an obvious limit process. To establish convergence you first prove total boundedness (in the context i just mentioned this involves the use of the relative entropy function or sometimes the boundedness of the increments). This automatically guarantees convergence along subsequences (and this is a consequence of a general Bolzano-Weirstrass theorem). It remains to show that each subsequence has a further subsubsequence that converge to the candidate limit process. This is usually clear intuitively and can be proved as a consequence of a general law of large numbers type result.

The second one, finding a fast converging subsequence which approximate the whole sequence well: this I would call a hard analysis method. You first have to find your fast convergence sequence: this means to do computation to show that the subsequence you picked does converge at a certain rate. Then you have to also show, via hard calculation, that the whole sequence doesn't wander off too much from this subsequence. Almost all `convergence almost surely' arguments that I know that don't use martingales are of this form. For example: law of the iterated logarithm, the strong law of large numbers. I would say this is one of the main methods of proving almost sure convergence (of random variables, processes).

Because of these differences I think it would be logical to have separate articles for them. Whether to have a further page like the one i proposed? My personal take is I always like classification, if there is an obvious way to classify arguments then I would do. These are different types of arguments typically used for different goals and appearing in different contexts, but they both use subsequences. There may be other methods using subsequences and to me it would be nice to have a page that lists them all together.

OK, why not have a look at

OK, why not have a look at what I've done and see what you think. I don't insist on keeping things like this.