real analysis - Prove if {$a_n$} $rightarrow$ $infty$, then {$a_n$} is not bounded above. Give an indirect proof.

The book I am using for my Advance Calculus course is Introduction to Analysis by Arthur Mattuck.

Prove if {$a_n$} $\rightarrow$ $\infty$, then {$a_n$} is not bounded above. Give an indirect proof.

This is my rough proof to this question. I was wondering if anybody can look over it and see if I made a mistake or if there is a simpler way of doing this problem. I want to thank you ahead of time it is greatly appreciated.So lets begin:

Proof: