determine if $f_n(x) = \sin(\frac x n)$ at $\mathbb{R}$ converges pointwise and uniformly

Question

I'm required to determine if $f_n(x) = \sin(\frac x n)$ at $ \mathbb{R}$ Converges Pointwise and uniformly.

Now, because my range is $ \mathbb{R}$ I'm leaning towards diverges, but im not quite sure how to go about proving any of the ways.

I know that since my function is Sin, its supremum would be 1, but im not sure how to use that information.

I would appreciate any advice on how to move forward with the proof.

Answer 1

If it converges uniformly and pointwise then the two limits will agree. For any $x$, $x/n \to 0$ and so since $\sin$ is continuous it will converge pointwise to $0$. For uniform convergence you would need that the supremum of $|\sin(x/n)|$ tends to 0, but this is always 1 for any value of $n$.