Pick out the sequences $\{f_n\}$ which are uniformly convergent.
(a) $f_n(x) = nxe^{−nx}$ on $(0,∞)$.
(b)$f_n(x) = x^n$ on $[0, 1]$.
(c)$f_n(x) = \frac{\sin(nx)}{\sqrt{n}}$ on $\mathbb{R}$.
(d) $f_n(x)=\frac{nx}{1 + nx}$ on $(0,1)$
(e) $f_n(x) = ∑_{n=1}^∞\frac{n\sin(nx)}{e^n}$ on $[0,\pi]$
(f) $f_n(x) = \frac{x^n}{1 + x^n}$ ; on $[0, 2]$
(g) $f_n(x) = \sin^nx$ on $[0,\pi/2)$
(h) $f_n(x) = (x^n/n)+1$ on $[0,1)$
(i) $f_n(x) = \frac{1}{1+(x-n)^2}$ on $(-∞,0)$
(j) $f_n(x) = \frac{1}{1+(x-n)^2}$ on $(0, ∞)$
(a) Here the function becomes $nx/e^{nx}$ which is tends to $0$ as $n$ tends to $∞$ so the function is uniformly continuous. I am not sure though.
(b) Here limit function is not continuous so not uniformly continuous
(c) true by same reason of (a) (d) false as $n$ tends to $∞$ the function tends to $1$ not zero.
(e) no idea (f) false as limit function is not continuous (g) true by M test (h) true by M test (i) no idea (j) no idea
Can somebody guide me properly please