(i) Suppose that $\left(f_{n}\right)$ is a decreasing sequence of continuous real-valued functions on a compact metric space $(X, d)$ which converges pointwise to 0 . By considering sets of the form $B_{n}=\left\{x: f_{n}(x)<\epsilon\right\}$, for $\epsilon>0$, or otherwise, show that $f_{n}$ converges uniformly to 0 .

Can the condition that $\left(f_{n}\right)$ is decreasing be dropped? Can the condition that $(X, d)$ is compact be dropped? Justify your answers.

(ii) Suppose that $k$ is a positive integer. Define polynomials $p_{n}$ recursively by

$$p_{0}=0, \quad p_{n+1}(t)=p_{n}(t)+\left(t-p_{n}^{k}(t)\right) / k$$

Show that $0 \leqslant p_{n}(t) \leqslant p_{n+1}(t) \leqslant t^{1 / k}$, for $t \in[0,1]$, and show that $p_{n}(t)$ converges to $t^{1 / k}$ uniformly on $[0,1]$.

[You may wish to use the identity $a^{k}-b^{k}=(a-b)\left(a^{k-1}+a^{k-2} b+\ldots+b^{k-1}\right)$.]

Suppose that $A$ is a closed subalgebra of the algebra $C(X)$ of continuous real-valued functions on a compact metric space $(X, d)$, equipped with the uniform norm, and suppose that $A$ has the property that for each $x \in X$ there exists $a \in A$ with $a(x) \neq 0$. Show that there exists $h \in A$ such that $0<h(x) \leqslant 1$ for all $x \in X$.

Show that $h^{1 / k} \in A$ for each positive integer $k$, and show that $A$ contains the constant functions.