Let $\ \varphi \, : \, V\rightarrow V\ $ be a linear transformation. Prove that $\ Im(\varphi \, \circ \varphi) \subseteq Im \,\varphi\ $ Let V be a vector space and $\ \varphi \, : \, V\rightarrow V\ $ be a linear transformation. Prove that:
$$\ Im(\varphi \, \circ \varphi) \subseteq Im \,\varphi\ $$
I am struggling to see what conditions I have to verify, in order to prove it.

Let $x\in Im(\varphi\circ\varphi)$ and note that $x=\varphi(\varphi(y))$ for some $y\in V$ by the definition of $Im(\varphi\circ\varphi)$. But then $x=\varphi(z)$ where $z=\varphi(y)$ and so $x\in Im(\varphi)$ by the definition of $Im(\varphi)$ as $z\in V$. It follows that $Im(\varphi\circ\varphi)\subseteq Im(\varphi)$.
