Recently I learned about the Bolzano-Weierstrass theorem. The theorem is the following:
In $\mathbb R$ every bounded sequence contains a convergent subsequence.
A sequence $a_n$ is bounded if $a_n \in [-C,C]$ for some $C$. The proof that I saw was doing a bisection of $[-C,C]$ into subintervals of decreasing length. I understand the proof but I found another proof that seems slightly less complicated but I don't know if it is correct. Please can someone read my proof and tell me if it is correct?
Proof: If $a_n$ is bounded then $A = \{a_n \mid n \in \mathbb N \}$ is bounded. By the axiom of completeness $a = \sup A$ exists. By the definition of $\sup$ for every $k$ there is $a_{n_k} \in A$ with $|a_{n_k} – a|<{1 \over k}$. The $a_{n_k}$ are a convergent subsequence of $a_n$.
Best Answer
My mistake was that I assumed there are infinitely many $a_n$ near the $\sup$ of the set. But this is not implied by the definition of $\sup$.