Proving the uniform convergence of $ f_n (x) = \sin(\frac{1}{n})\cos(x)$

Question

I need to find the pointwise limit and determine the uniform convergence of the sequence of functions $ f_n (x) = \sin(\frac{1}{n})\cos(x)$ where $f$ maps from reals to reals.

I got the pointwise limit to be $0$, but I am unsure how to prove the uniform convergence. I believe it starts with bounding this function -- upper bound is $1$ and lower bound is $-1$. I'm not sure where to go from there.

Answer 1

Hint: $$|\sin(x)| \leqslant |x|$$ So what can you say about $$||f_n-0||_u=\sup_x\left|\sin\left(\frac{1}{n}\right)\cos(x)\right|\text{ ?}$$