Does the sequence $(f_n)$ of functions $f_n:(0,\infty)\to\mathbb R$
$$f_n(x) = \begin{cases} 0 & x \in (0,\frac1{n+1}) \\ \sin^2 \frac\pi x& x \in [\frac1{n+1},\frac1n]\\ 0 & x \in (\frac1n, \infty)\end{cases}$$
converge uniformly?
I am fairly new to this and have no idea how to go about this so any help would be great.
Hint
The pointwise limit of $f_n$ (note that $n$ is a parameter) is $$\lim_{n\to\infty} f_n(x) = 0 \qquad \forall\ x\in (0,\infty)$$ So the uniform limit is $0$ if it exists. Now see that $$\sup_{\theta\in [n\pi, (n+1)\pi]} |\sin\theta| = 1$$ what do you conclude for $\|f_n - 0\|_\infty$? What does this mean for $$\lim_{n\to\infty} \|f_n\|_\infty$$