What is meant by a "bump function" on $\mathbb{R}^{n}$ ? If $U$ is an open subset of a manifold $M$, prove that there is a bump function on $M$ with support contained in $U$.

Prove the following.

(i) Given an open covering $\mathcal{U}$ of a compact manifold $M$, there is a partition of unity on $M$ subordinate to $\mathcal{U}$.

(ii) Every compact manifold may be embedded in some Euclidean space.