Sequence converging to supremum of a function

Question

Let $A$ be some bounded set and $f:A\rightarrow\mathbb{R}$ be some upper bounded, continuous function. Since $f$ is bounded, we know that $M:=\sup_{x\in A}\; f(x)<\infty$ exists. Why can we always find an increasing sequence $(a_n)_{n\in \mathbb{N}}\;\subset A$ such that $f(a_n)$ converges to $M$?