Show that there exist no retraction from $\mathbb{RP}^n$ to $\mathbb{RP}^k$ if $n>k$.

Question

I am trying this problem from Hatcher's algebraic topology book (4.2.1). If $r:X\rightarrow A$ is retraction then I know that this induces injective map in the fundamental group level through inclusion $A\subset X$. So from fundamental group I am not able to solve this problem. I don't know role of higher homotopy groups in retraction problems. Could anyone please give me some hint? Thanks.