Prove that this space is compact/not compact by using Arzelà–Ascoli theorem

Question

I need to tell if this space/set is compact in $C[0,1]$

$x_n(t) = t^n, n ∈ N$

Following Arzelà–Ascoli theorem, the set is compact when it has Uniform boundedness and Equicontinuity, is it correct?

So Uniform boundedness, in my case:

$Ǝ M: ∀φ(t)∈Q=>|φ(t)|≤M,∀t∈[0,1]$

$|X_n(t)|=|t^n|≤1$

But I'm really confused whether it's correct or not, or how to solve this overall

Answer 1

The space is not compact because every subsequence of the sequence $(x_n)_{n\in\mathbb N}$ converges pointwise to 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 each $x_n$ is continuous. Therefore, the convergence cannot be uniform. In other words, every subsequence of the sequence $(x_n)_{n\in\mathbb N}$ diverges with respect to the $\sup$ metric.