I was wondering whether someone could provide some feedback on my proof. Thank you!
Problem. Let $(a_{n})_{n\in\mathbb{N}}$ be an increasing and bounded sequence. Prove that $a_{n}\rightarrow\sup{(a_{n})}$.
Suppose $(a_{n})_{n\in\mathbb{N}}$ is increasing and bounded, and suppose that $L$ is the least upper bound. Then $\forall n\in\mathbb{N}, a_{n}\in[a_{1},L)\implies a_{1}\le a_{n}\le L$. Let $\epsilon>0$ be given. Then $L-\epsilon$ is not a upper bound, so there exists $k\in\mathbb{Z}$ such that $L-\epsilon<a_{k}$. Since $a_{n}$ is an increasing sequence, then for all $n\ge k$, $|a_{n}-L|<\epsilon. \blacksquare$