A convergent sequence is bounded: Proof Verification

Question

May you please verify my proof. I feel like I am not being rigorous, and assuming that the infimum and the supremum are finite.

Theorem: A convergent sequence $(a_n)_{n\in \Bbb N}$ is bounded.

Proof:

Let $l\in \Bbb R$ such that $a_n \rightarrow l$.

Then $\forall \epsilon>0\,,\exists n_0\in \Bbb N;\forall n \in \Bbb N,\, n\ge n_0 \implies|a_n-l|<\epsilon $

Let $\epsilon=Max\{|sup((a_n)_{n\in \Bbb N})|, \, |inf((a_n)_{n\in \Bbb N})|\}$,

Then $|a_n-l|<\epsilon\,\,\,\,\forall n\in \Bbb N\,\,$

Then $l-\epsilon<a_n<l+\epsilon\,\,\, \forall n\in \Bbb N$

So $(a_n)_{n\in \Bbb N}$ is bounded.

I know the classical proof of this, but I was wondering if this workes too! Any helps is greately appreciated!

Answer 1

Assume it is not bounded. Then $$(\forall n\in \Bbb N) \;\; (\exists \phi (n) \in \Bbb N)\;:$$ $$|a_{\phi (n)}|>n $$

thus $b_n=a_{\phi (n)} $ will be a divergent subsequence of a convergent one.