Does a monotonically increasing function attain a maximum on a compact subset of $\mathbb{R}$? Prove that it does, or give an example against.

Question

Does a monotonically increasing function attain a maximum on a compact subset of $\mathbb{R}$?

I have to either prove that it does, or give an example that shows that it doesnt.

My attempt at it is that since for any $a>b, \Rightarrow f(a) \geq f(b)$, then the function will attain a max, it follows directly that on the interval $[c,d]$ the function will attain its max at $d$.

Is this correct? How can I prove it? Or is it wrong? If so, give an example against it.

Answer 1

If $K$ is a non-empty compact subset of $\mathbb R$, then it has a maximum $m$. And, for every $x\in K$, since $x\leqslant m$, $f(x)\leqslant f(m)$. So, the maximum is attained at $m$.