uniformly convergence

Question

suppose $f_n(x)$ is a sequence of complex functions which converges uniformly to $f(x)$,$g_n(x)$ is a sequence of complex functions such that $|f_n(x)-g_n(x)|\to 0$.Can we conclude that $g_n(x)$ is uniformly convergent to $f(x)$?

Answer 1

Unless $\lvert{f_n(x) - g_n(x)}\rvert \rightarrow 0$ is uniform, no. For instance, take $f_n(x) \equiv 0$ and $f \equiv 0$. Then $f_n \rightarrow f$ uniformly on $(0, 1) \subset \mathbb{R} \subset \mathbb{C}$. Take $g_n(x) = x^n$. Then $\lvert{g_n(x)\rvert} \rightarrow 0$ for all $x \in (0, 1)$ pointwise but not uniformly.

If $\lvert{f_n(x) - g_n(x)\rvert} \rightarrow 0$ uniformly, then your statement is true as $$\lvert{g_n(x) - f(x)\rvert} \leq \lvert{g_n(x) - f_n(x)\rvert} + \lvert{f_n(x) - f(x)}\rvert$$ and you can bound both terms uniformly by assumption.