Need help on monotone increasing sequence

Question

Let $\{a_n\}_{n\in\mathbb N}$ be a monotone increasing sequence where there exists $N\in\mathbb N$ such that for any $n, m \in\mathbb N$ with $n, m \geq N$ we have that $|a_n − a_m| \leq 2$. Show that $\{a_n\}_{n\in\mathbb N}$ is convergent.

Answer 1

So \begin{align*} |a_{n}-a_{N}|\leq 2,~~~~n\geq N, \end{align*} and hence \begin{align*} a_{n}-a_{N}\leq 2,~~~~n\geq N, \end{align*} the sequence is bounded above by $\max\{a_{N}+2,a_{1},...,a_{N-1}\}$. Monotone increasing which is bounded above is convergent.

Answer 2

We prove that the sequence is bounded and as you know a monotone bounded sequence is convergent.

According to the assumption we have $$|a_n-a_N |\le 2$$ for $n\ge N$

Thus for all $n\ge N$ we have $$|a_n|\le |a_N|+2$$

Define $$M= max \{ |a_1|, |a_2|,..., |a_{N-1}|, 2+|a_N|\}$$

Then all the terms of your sequence are bounded above by $M$ thus your sequence is a bounded monotone sequence thus it is convergent.