Proof of Extreme Value Theorem on a Compact Interval

Let $f:[a,b]\to\mathbb R$ be continuous. Since $[a,b]$ is compact, its image $f([a,b])$ is compact in $\mathbb R$, hence closed and bounded. Therefore $M:=\sup f([a,b])$ and $m:=\inf f([a,b])$ are finite and belong to $f([a,b])$. So there exist $x_M,x_m\in[a,b]$ with $f(x_M)=M$ and $f(x_m)=m$. Thus $f$ attains both a maximum and a minimum on $[a,b]$.