I want to prove that
If $E$ is a Vector Space, $F,G$ are subspaces of $E$ and if $a+F\subseteq G$, for some $a \in E$, then $F \subseteq G$.
I tried to prove that $a+F$ is subspace of $E$ but I think that is false. Here my attempt:
Suppose that $x\in F$, then $x+a\in a+F$, note that $a+F$ is subspace (In this part I think that is wrong) then $x+a+(-a)\in a+F$ but $a+F\subseteq G$ thus $x+a+(-a)=x\in G$
Since $a + F \subset G$, and $F$ is a subspace (and thus must contain $0$), $a \in G$. Let $x \in F$. We then have that $$x = a + (x-a) = (a+x) -a \in G,$$ since $a+x\in G$ and $-a \in G$.