Uniformly convergent sequence of function takes uniformly continuous functions to uniformly continuous function or not?

Question

I know that

Let $X \subset \mathbb R$ and $f_n:X \to \mathbb R$ be continuous, if $f_n \to f$ uniformly then $f$ is continuous.

My question is:

Is this true for Uniformly continuous functions?

I suppose it is not.Help me providing a counter example.

Answer 1

A uniformly convergent sequence of uniformly continuous functions converges to a uniformly continuous function.

Let $\varepsilon > 0$ be arbitrary. By uniform convergence we can find an $n$ so that $\|f - f_n\|_\infty < \varepsilon/3$. Now choose $\delta > 0$ in such a way that $|x - y| < \delta \implies |f_n(x) - f_n(y)| < \varepsilon/3$. This implies that for any $x, y$ with $|x - y| < \delta$ we have

$$|f(x) - f(y)| \le |f(x) - f_n(x)| + |f_n(x) - f_n(y)| + |f_n(y) - f(y)| \le \varepsilon/3 + \varepsilon/3 + \varepsilon/3 = \varepsilon,$$

i.e. $f$ is uniformly continuous.