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.

More generally, for linear maps $U \xrightarrow{\phi} V \xrightarrow{\psi} W$, we have $\mathrm{Im}(\psi \circ \phi) \subseteq \mathrm{Im}(\psi)$.
Proof: $\mathrm{Im}(\psi \circ \phi) = (\psi \circ \phi)(U)=\psi(\phi(U)) \subseteq \psi(V)=\mathrm{Im}(\psi)$.
