Where can I find references to proofs/can anyone supply me a quick proof of the following facts?

• If the $n$-dimensional manifold $M$ can be immersed in $\mathbb{R}^{n+1}$, then each $w_i(M)$ is equal to the $i$-fold cup product $w_1(M)^i$.
• If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, then $n$ must be of the form $2^r - 1$ or $2^r - 2$.