A question regarding the Bolzano Weierstrass Theorem

Question

This is probably a very stupid/absurd question....

The theorem states that every bounded real sequence has a convergent subsequence. We further know that every sequence of real numbers has a monotone subsequence.

My question is, why not simply use the fact that a bounded monotone sequence is convergent to prove the theorem in a simpler way?

Indeed the fact is used in the Cantor Intersection Theorem, but is the 'fact' alone not enough to prove BW theorem? Am I missing something?

Answer 1

There is nothing stupid or absurd in that question. Actually, some textbooks prove the Bolzano-Weierstrass theorem along this line. But it is not trivial or obvious that every sequence of real numbers has a monotone subsequence.

Answer 2

We can prove the Bolzano-Weierstrass theorem this way. First, we prove that every sequence has a monotone subsequence, and from this it follows that every bounded sequence has a convergent subsequence, by the Monotone Convergence theorem.

Here’s a proof sketch of the first part:

Let $(a_n)$ be a real sequence.

We say that a term $a_i$ in the sequence is dominant if $j > i \implies a_j \leq a_i$.

There are two possibilities for the set of all dominant terms.

1) The set is infinite. In this case, the subsequence formed from all the terms in this set is a decreasing sequence.

2) The set is finite. In this case, there must be a point in the sequence after which all terms are not dominant. So take a term $a_i$ after this point. Now, since $j > i \implies a_j \leq a_i$ for $a_i$ dominant we can find a term $a_j$ in the sequence that is greater than or equal to $a_i$. Iterate this process, and the subsequence formed will be an increasing sequence.

In either case, we have a monotone subsequence.