An application of Uniform Boundedness Theorem

Question

I did this problem before

"Let $X$ be a compact Hausdorff space and assume that $C(X)$ is equipped with a norm $||.||$ with which this is a Banach space. For each $x\in X$, define $\lambda_x:C(X)\to \mathbb{R}, f \mapsto f(x)$. Prove that $\sup_x ||\lambda_x||$ is bounded."

What I did: Fix $f\in C(X)$. Then we observe that $\sup_x |\lambda_x(f)|=\sup_x |f(x)|=||f||_{\infty}<\infty$. Hence by the Uniform Boundedness Principle, we deduce $\sup_x ||\lambda_x||$ is bounded.

This answer was marked correct. But now I read this, why is $||f||_{\infty}<\infty$?

Sorry if this is a dumb question.

Answer 1

Because every continuous function from a compact space into $\Bbb R$ is bounded.