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.

Recall that 
$$\mathrm{Im}(\varphi)=\{\varphi(x)\quad|\quad x\in V\}$$
Now take $y\in \mathrm{Im}(\varphi\circ \varphi)$ then there's $x\in V$ such that
$$y=\varphi\circ \varphi(x)= \varphi( \underbrace{\varphi(x)}_{z\in V})=\varphi(z)\in \mathrm{Im}(\varphi)$$
so we have
$$\ Im(\varphi \, \circ \varphi) \subseteq Im \,\varphi\ $$
