Does this sequence of functions have a converging subsequence?

Question

Define $(f_n)_{n\geq1}$ such that $f_n=sin(x+n)$, where this sequence is defined on $\mathbb{R}$. I know that Arzeli-Ascoli cannot be helpful here since $\mathbb{R}$ is not compact, so is there a way of testing the existence of converging subsequence?

Answer 1

If you are talking about pointwise convergence the answer is YES: For each $N$ there is a subsequence which converges pointwise (even uniformly) on $[-N,N]$. Hence we can take a diagonal subsequence which converges for all $x$.

There is no subsequence which converges uniformly: if $|sin (nx)-\sin (mx)| <1$ for all $n , m >N$ for all $x$ we can get a contradiction by taking $x=\frac {(2k+1) \pi} {2n}$ and $m=2n$ with $k$ large enough.