Set with elements of the form $𝑓_𝑛:[0,1] →ℝ$ defined as $𝑓_𝑛 (𝑥) = 𝑥^ 𝑛~ \text{for}~\text{ all}~ 𝑥∈ [0, 1].$ is closed.

Let $$𝐶[0,1] = \{ 𝑓: [0, 1] →ℝ∶𝑓 ~\text{is}~\text{ continuous}\}$$ and$$d_{∞}(𝑓, 𝑔) = sup\{ |𝑓(𝑥) −𝑔(𝑥)|: 𝑥∈ [0, 1]\} $$for $𝑓, 𝑔∈𝐶[0,1].~\text{For}~\text{ each}~ 𝑛∈ℕ$, define $𝑓_𝑛:[0,1] →ℝ$ by $𝑓_𝑛(𝑥) = 𝑥^𝑛~\text{for}~\text{ all}~ 𝑥∈ [0, 1].$Let $𝑃 = {𝑓_𝑛: 𝑛∈ℕ}.$

I want to know , $P$ is closed or not.

Since only constant sequence ( $(x^k)_n$ for fixed k) are convergent sequences here and limit is also in $P$ hence $P$ is closed.

Is it correct?

Also can we say it is not open?