Continuous function/Closed bounded interval/Maximum is attained/Fact

Theorem on attained maximum

Let $[a,b]\subseteq ℝ$ be a closed bounded interval, and let

$f:[a,b]⟶ℝ$

denote a continuous function.

Then there exists some

$x\in [a,b]$ such that

$f(x)\ge f(x^{\prime })\text{ for all }{x}^{\prime }\in [a,b].$

This means that the function attains its maximum (and its minimum).

Write a proof