Define an immersion and an embedding of one manifold in another. State a necessary and sufficient condition for an immersion to be an embedding and prove its necessity.

Assuming the existence of "bump functions" on Euclidean spaces, state and prove a version of Whitney's embedding theorem.

Deduce that $\mathbb{R}^{n}$ embeds in $\mathbb{R}^{(n+1)^{2}}$.