The question is to check whether the sequences are uniformly convergent or not:
$f_n(x)=Sin^nx$ on $[0,\frac{\pi}{2})$
$\frac{1}{1+(x-n)^2}$ on $(-\infty.0)$
$\frac{1}{1+(x-n)^2}$ on $(0.\infty)$
For $(1)$
$f_n(x)\to0$. Now $sup[f_n(x)-0]$=$1$(It never attains $1$ but it is not mandatory for the supremum to lie in the set). Hence $(1)$ doesn't converge uniformly.
For $(2)$
$f_n(x)\to0$
$\frac{1}{1+(x-n)^2}\le\frac{1}{1+n^2}\le\frac{1}{n^2}\le\epsilon$. Here $n$ bdepends only on $\epsilon$. Hence uniformly convergent.
For $(3)$
$f_n(x)\to0$
$sup[f_n(x)]=1$, which it attains at $x=n$. Hence not uniformly convergent.
Is there any better way of showing these?? By sequential criteria?
