Suppose that $f:\Bbb{R}\to\Bbb{R}$ is uniformly continuous. Let $f_n(x)=f(x+1/n)$.
a) Prove that $f_n$ converges uniformly to $f$ on $\Bbb{R}$
b) Does this remain true if $f$ id just continuous? Prove it or provide a counterexample.
My attempt:-
For a, since $f$ is uniformly continuous, $f(x+1/n)$ exists and $\limsup\left|f_n-f\right|=\limsup\left|f(x+1/n)-f(x)\right|=0$. Therefore $f_n\to f$ uniformly on $\Bbb{R}$.