I want to show if the following sequence of functions converges pointwise and converges uniformly on R? $ f_n(x) = \sin(x/n) $
Here's what I did: -Fix $\displaystyle x\in R, \lim_{n\rightarrow\infty}f_n(x) = \lim_{n\rightarrow \infty} \sin(n/x) = 0 $. so $f_n $ converges to f s.t. f(x) = 0 for all x in R, point wisely. Now for uniform convergence: $\|f_n - f\|_{\sup} = \sup_R|f_n - f|= \sup_R|\sin(x/n)-0| = \sup_R|\sin(x/n)|\rightarrow0 $ as n$\rightarrow\infty$, so it's uniformly convergent. Is this correct? and should I right anything more for the uniform convergence?