Image of a convergent sequence in an increasing function

Question

Suppose I have a function $f:[a,b) \to \mathbb{R}$ that is increasing on its domain and a sequence $a_n \subset [a,b)$ such that $a_n \to b$ and $f(a_n)\to l\in \mathbb{R}$.

How would I go about showing that $l=sup(im \ f)$?

Any hints would be well appreciated, not full answers.

Answer 1

Let $x_0 \in [a,b)$ be arbitrary. We can always pick $N$ s.t. $n > N \implies a_n \in (x_0,b) \implies f(a_n) \geq f(x_0)$.

Answer 2

For any $x\in [a,b)$, you have $a_n$ such that $x< a_n$, and hence $f(x) \le f(a_n) \le l$.