Question

Let {an}∞n=1 be a non-decreasing (resp. non-increasing) sequence which converges to a. Then prove that an ≤ a (resp. a ≤ an) for every n ∈ N

Answer 1

Let $(a_n)_{n=1}^{\infty}$ be a non-decreasing (resp. non-increasing) sequence which converges to

$a$, so we have that

$\lim \limits_{n \to \infty} a_n = a$ $... (*)$

Also, $a_n \leq a_{n+1}$ (resp. $a_n \geq a_{n+1}) \forall n \geq 1$

So we have from $(*)$ that $\lim \limits_{n \to \infty} a_n \leq a$

(resp. $\lim \limits_{n \to \infty} a_n \geq a$ ).

And since $(a_n)_{n=1}^{\infty}$ is non-decreasing (resp. non-increasing), we have that $a_n \leq a$ (resp. $a_n \geq a$ ) As desired