Which one of the given sequences of functions is uniformly convergent $?$ Which one  of  the  given  sequences  of  functions  is  uniformly  convergent $?$
$$A.\ \ f_n(x)=x^n;x\in[0,1].$$
$$B.\ \ f_n(x)=1-x^n;x\in\left[{1\over2},1\right].$$
$$C.\ \ f_n(x)={{1}\over{1+nx^2}};x\in\left[0,{1\over 2}\right].$$
$$D.\ \ f_n(x)={{1}\over{1+nx^2}};x\in\left[{1\over2},1\right].$$
I think option $D.$ is correct . For if we take $\lim_{n\rightarrow \infty}f_n=f$  then for 
$A$ $f=1$ at $1$ and $0$ elsewhere.
$B$ $f=0$ at $1$ and $1$ elsewhere.
$C$ $f=1$ at $1$ and $0$ elsewhere.
Did I got things right $?$ 
Thank you.

Noting that the uniform limit of continuous functions must be continuous too, only $D$ is possible (as your (correct) calculations have shown). 
For the reason why D is continuous on the give interval, you can use Dini's theorem as is indicated in the previous answer. Or more directly, note that 
$$|f_n(x)-f_m(x)|=\frac1{(1+mx^2)(1+nx^2)}|m-n|x^2\le\frac1{16mn}|m-n|\le \frac1{16mn}(m+n)\le \frac1{16}(\frac1m+\frac1n)$$
Then use Cauchy's criterion for uniform convergence. 
