1. ## Compact Sets.

Let $\displaystyle (X,d_{x})$ be a continuous function. Suppose that $\displaystyle K\subset X$ is compact and that $\displaystyle A\subset\ \mathbb{R}^k$ is compact.
Prove that $\displaystyle f(K)\bigcap A\ =\ \emptyset$ implies dist$\displaystyle (f(K),A)>0$

It's analysis and there isn't an extra class for that unlike the other modules (even though it's considered the hardest).
Let $\displaystyle (X,d_{x})$ be a continuous function. Suppose that $\displaystyle K\subset X$ is compact and that $\displaystyle A\subset\ \mathbb{R}^k$ is compact.
Prove that $\displaystyle f(K)\bigcap A\ =\ \emptyset$ implies dist$\displaystyle (f(K),A)>0$
You know that $\displaystyle f(K)$ is compact, and so $\displaystyle f(K)\times A$ is compact, so the mapping $\displaystyle d:f(K)\times A\to\mathbb{R}$ obtains a minimum value $\displaystyle d(f(k_0),a_0)$ for some $\displaystyle (f(k_0),a_0)\in f(K)\times A$. Clearly though $\displaystyle d(f(k_0),a_0)>0$ otherwise $\displaystyle f(k_0)=a_0$ which contradicts $\displaystyle f(K)\cap A=\varnothing$.