Let $n \leq k$ positive integers show that the natural embedding $\mathbb{R}^{n+1} \hookrightarrow \mathbb{R}^{m+1}$ induces an embedding $\mathbb{RP}^{n} \hookrightarrow \mathbb{RP}^{m}$
This is my atemp: Let $j$ the embedding from $\mathbb{R}^{n+1}$ to $\mathbb{R}^{m+1}$ then the restriction of $j$ in $\mathbb{S}^{n}$ induces a embeding $l$ from $\mathbb{S}^{n}$ in $\mathbb{S}^{m}$, i know that $\mathbb{RP}^{n}=\mathbb{S}^n / \sim $ and $\mathbb{RP}^{m}=\mathbb{S}^m / \sim $, with $ \sim$ the antipolar relation.
I try to define $J: \mathbb{RP}^n \rightarrow \mathbb{RP}^{m}$ by $J([x])=[l(x)]$ but i cant conclude that $J$ is one to one because if $J([x])=J([y]) \Rightarrow [l(x)]=[l(y)]$ and i have problems in these part.
Maybe i think to is a better idea work with the characterization of $\mathbb{RP}^n$ given by the quotient $(\mathbb{R}^{n+1} \setminus {0}) / \sim' $ with $\sim'$ the relation given by $x \sim' \lambda x, \forall \lambda \in \mathbb{R}$, with $\lambda \neq 0$
Any hint or help i will be very grateful
