Proofs from Sequences and limits

For a non-decreasing sequence bounded above, let $L=\sup\{a_n:n\in\mathbb N\}$. Given $\varepsilon>0$, some term $a_N$ is greater than $L-\varepsilon$; otherwise $L-\varepsilon$ would be a smaller upper bound. Monotonicity gives $L-\varepsilon<a_N\le a_n\le L$ for all $n\ge N$, so $a_n\to L$. The non-increasing case uses the infimum in the same way.