Is {$fn$} and {$gn$} is uniformly convergent?

Question

Consider two sequence $f_n$ and $g_n$ of functions where $f_n:[0,1] \to \mathbb{R}$ and $g_n: \mathbb{R} \to \mathbb{R}$ which are defined by

$f_n(x) = x^n$ and
$ g_n(x) = \begin{cases} cos(x-n)\frac{π}{2} &\mbox{if } x ∈ [n-1,n+1] \\ 0 & \text{otherwise}. \end{cases} $

Then

1) Neither $f_n$ nor $g_n$ is uniformly convergent

2) $f_n$ is not uniformly convergent but $g_n$ is

3) $g_n$ is not uniformly convergent but $f_n$ is

4) Both $f_n$ and $g_n$ are uniformly convergent...

I think both $f_n$ and $g_n$ are pointwise convergent so they will not uniformly convergent. Therefore option 1 is correct that is neither $f_n$ nor $g_n$ is uniformly convergent.

Is my answer is correct or not? Please help me or tell me the solution,,, I would be more thankful.... thanks in advance

Answer 1

Neither of them converges uniformly:

1. The pointwise limit of $(f_n)_{n\in\mathbb{N}}$ is the function$$\begin{array}{rccc}f\colon&[0,1]&\longrightarrow&\mathbb{R}\\&x&\mapsto&\begin{cases}0&\text{ if }x<1\\1&\text{ otherwise.}\end{cases}\end{array}$$which is discontinuous. But a sequence of continuous functions cannot converge uniformly to a discontinuous function.
2. The pointwise limit of $(g_n)_{n\in\mathbb{N}}$ is the null function but, for each $n\in\mathbb N$, there is a real number $x$ (namely, $n$) such that $g_n(x)=1$. Therefore, the convergence cannot be uniform.