If I have a real sequence $(x_n)_{n \in \mathbb{N}}$ which is monotone increasing, i.e. $x_n \leq x_{n + 1}$ for all $n \in \mathbb{N}$, I think it is true that $$\sup\{x_n : n \in \mathbb{N}\} = \lim_{n \to \infty}x_n$$ Has anyone a hint for me how I could prove this?
[Math] Limit of monotone increasing sequence is supremum
real-analysis
Best Answer
Hint: By the definition of supremum, given $\epsilon > 0$ , $\sup x_n - \epsilon$ is not an upper bound, so there is some $M$ such that $x_M > \sup x_n - \epsilon$. Now, if $N > M$, then $x_N > \sup x_n - \epsilon$. This should help.