This is derived of other question where my proof of the Bolzano theorem is as follows
Proof: Suppose that $(a_n)$ is a bounded sequence then we have to show that it has a convergent subsequence. Since it is bounded then it has a finite limit point $c$, for example $c= \limsup a_n$.
Let define recursively the following sequence of natural number, $n_0=0$ and $n_i=\min(\{n> n_{i-1}:|a_n-c|\le 1/i\})$. Since $c$ is a limit point then the set is non-empty and the $\min$ is well-defined. Then $(a_{n_i})$ is a subsequence of $a_n$ and since $|a_{n_i}-c|\le 1/i$ for all $i\ge 1$ then the sequence converges to $c$, i.e., $(a_{n_i})\rightarrow c$. $\square$
Is this redundant? I think it is correct and also I don't think that I'm using what must be proved in the proof - indeed, a pretty similar approach is using in my book of analysis. Am I right or Am I totally off track?
[Definition of limit point of a sequence (which I'm know): For all $\varepsilon >0$ and for all $N$ there is a $n\ge N$ such that $|a_n-c|\le \varepsilon$]
A few "fundamental" theorems are equivalent: Cantor's nested interval theory, the Bolzano Weierestrass theorem, completeness and the monotone convergence theorem. Assume one, and make sure you prove the others with this!