Proving that $x_n(t)=\sin\left(n\frac{(t-n^2)}{n+1}\right)$ has subsequence that converges

Question

I'm having some trouble with the following exercise:

Let $x_n:[a,b]\to\mathbb R$ be a sequence of functions given by$$x_n(t)=\sin\left(n\frac{(t-n^2)}{n+1}\right)$$ Prove that $x_n$ has a subsequence that converges uniformly.

I'm thinking about using Arzelà–Ascoli theorem: Let $H=\{x_n,n\in\mathbb N\}\subset\mathcal C([a,b],\mathbb R)$. I was able to prove that $H(x):=\{f(x),f\in H\}$ is relatively compact, but now I'm having some trouble proving that $H$ is equicontinuous. How can this be done?

Answer 1

It is easy to prove by the MVT that $|\sin(x)-\sin(y)|\leq |x-y|$. Apply this inequality to enable bounding of $|x_n(t)-x_n(t_0)|$. This should give equicontinuity.