Uniform convergence to f(x) = 0

Question

I want to prove that $$f_n(x) = \frac{\sin(n^2 x)}{n}$$ converges to $f(x) = 0$. It's easy to see that $$\lim_{n\to\infty} \frac{\sin(n^2 x)}{n}= 0$$ but how can I prove the uniforme convergence ?

Answer 1

Use the fact that$$(\forall n\in\mathbb N)(\forall x\in\mathbb R):\left\lvert\frac{\sin(n^2x)}n\right\rvert\leqslant\frac1n.$$

Answer 2

The convergence is uniform since$$\left|\frac{\sin (n^2x)}{n}\right| \le\frac1n.$$

The upper bound is independent from $x$.