Suppose ${0<r<1}$. For each k $\in$ $\mathbb{N}$, define F$_{k}$ $\in$ C$\bigl($[-r,r]$\bigr)$ by F$_{k}$(x) = $\sum_{n=1}^k$ x$^{n}$.
Show (F$_{k}$) converges uniformly to some continuous function f $\in$ C$\bigl($[-r,r]$\bigr)$.
Somehow this has something to do with compact sets and metric spaces, as that was the section this homework was posted in. I'm not sure how they relate, however.
$$F_k(x)=\sum_{n=1}^kx^n = \frac{x(1-x^k)}{1-x}$$ converges to $$F(x) = \frac{x}{1-x}$$
Then $$\left|F_k(x)-F(x)\right| = \left|\sum_{n=k+1}^\infty x^n\right| = |x^k F(x) | \le r^k F(r)$$
The last value tends to $0$ when $k \to \infty$ and is independent of $x$. Uniform convergence follows.