Point wise and uniform convergence of sequences of functions

Question

Analyze the uniform convergence of the following sequences of functions:

1. $x+(1/n)$. What can we conclude about $(x+(1/n))^2$?

2. $1/(1+x)^n$ in $[0,1]$. Also, study the point wise convergence of this sequence in the same interval.

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I have done point 1. It converges uniformly to $x$. Regarding $(x+(1/n))^2$, I concluded it doesn't converge uniformly. Is this correct?

Now, I honestly don't know how to tackle point 2. Any help would be greatly appreciated. Thanks in advance!

Answer 1

Concerning the first sequence, you are right: $\left(\left(x+\frac1n\right)^2\right)_{n\in\Bbb N}$ converges pointwise, but not uniformly, to $x^2$ (on $\Bbb R$).

The second sequence converges pointwise to$$\begin{array}{rccc}f\colon&[0,1]&\longrightarrow&\Bbb R\\&x&\mapsto&\begin{cases}1&\text{ if }x=0\\0&\text{ otherwise.}\end{cases}\end{array}$$Since $f$ is discontinuous and each of the functions from the sequence is continuous, the convergence is not uniform.