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# Class Contents

##
Subsequences

In the following examples, we take the sequence:

Example:

Example:

Example:

Example:

Example:

One important reason subsequences are useful is because very often even when a sequence does not converge itself, it does have subsequences which converge.

Example:

On the other hand, if a sequence *does* converge then all its
subsequences also converge, and to the same thing.

Remark:

Although it seems a pretty unsurprising result, this last proposition
does give us a useful criterion for establishing that a given sequence
does *not * converge. All one has to do is to identify two
subsequences which converge to different limits. Since we have far more
results at our disposal which *do* prove convergence than we have
which disprove it, this is oftentimes much easier.

Example:

The following theorem shows that, at least in the real numbers, many sequences have convergent subsequences. So this behavior is very common indeed.

Remark:

This proof of the Bolzano-Weierstrass theorem uses ideas based on Ramsey's Theorem from combinatorics. Indeed the key step of the proof, showing that every sequence of real numbers has a monotonic subsequence can be proved using Ramsey's Theorem.

We can use Ramsey's theorem to prove again that

Following on the Bolzano-Weierstrass Theorem, we can use obtain a useful property of closed and bounded subsets of the real numbers:

This property is studied in more detail in a future section.