Let $D\subset R$ and for each $n\in N, fn : D \to R$ is continuous on $D$. If the sequence $(fn)$ be uniformly convergent on D to a function $f$, then $f$ is continuous on $D$
From my post here I am allowed to interchange limits
If $c\in D$, then $$\lim_{x\to c} f(x) = \lim_{x\to c} \lim_{n\to \infty} f_n(x) = \lim_{n\to \infty} \lim_{x\to c} f_n(x) = \lim_{n\to \infty}f_n(c) = f(c)$$
so this proves $f$ is continuous on $D$?
Is this correct? I have posted this method because I found it relatively simpler.