I'm stuck on this proof, can anyone help?
The first is pretty straight forward. In general, to prove "$\displaystyle A\subset B$" you show: "if y is a member of A then y is a member or B".
Here you want to prove: $\displaystyle Im(S+ T)\subset Im(S)+ Im(T)$
If y is in Im(S+ T) then y= (S+ T)x for some x in V. But "(S+ T)x" is defined as Sx+ Tx so we have y= Sx+ Tx. Let $\displaystyle y_1= Sx$ and $\displaystyle y_2= Tx$. Then $\displaystyle y_1$ is in Im(S), $\displaystyle y_2$ is in Im(T), and $\displaystyle y= y_1+ y_2$. Therefore, y is in Im(S)+ Im(T). The rest of that should be straight forward.
The second may be even simpler. Saying "ST= 0" mean (ST)v= 0 for all v. But that is the same as S(Tv)= 0 so that Tv is in the kernel of S for all v.