TITLE: uniform convergence of the composition of two sequence of functions
QUESTION [3 upvotes]: Let $(f_{n}:\mathbb{R}\to\mathbb{R})$ be a sequence of continuous functions with uniform limit $f:\mathbb{R}\to\mathbb{R}$. Now, for each $n\in\mathbb{N}$, define $g_{n}:\mathbb{R}\to\mathbb{R}$ by $$g_{n}(x)=f_{n}\left(x+\frac{1}{n}\right).$$
I proved that $g_{n}\to f$ pointwise. Is it true that $g_{n}\to f$ uniformly?
I don't think so, but i failed to create the counterexample.
Give some advice! Thank you!

REPLY [6 votes]: Not even true in a simple case. Take $f_n(x)=x^2$ then $$|g_n(x)-f(x)|=\left|2\frac{x}{n}+\frac{1}{n^2}\right|$$
For $n>1$ one has:
$$\Vert g_n(x)-f(x)\Vert_\infty\geq \left|2\frac{n^2}{n}+\frac{1}{n^2}\right|\geq 2n >2 $$
So uniform convergence can be forgotten.
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