Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function/Corollary

Theorem

Let $S \subseteq \R$.

Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$ converging uniformly to $f : S \to \R$.

Let $f_n$ be continuous for all $n \in \N$.

Then $f$ is continuous.

Proof

Let $x \in S$.

As $f_n$ is continuous at $x$ and $f_n \to f$ uniformly, we have by Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function:

$f$ is continuous at $x$.

As $x \in S$ was arbitrary:

$f$ is continuous at all $x \in S$.

So $f$ is continuous.

$\blacksquare$