The union of two subspaces is not alwys a subspace [duplicate]
Solution 1:
Assume not. Then there is an element $x \in F$ which isn't in $G$ and there is a $y \in G$ which isn't in $F$. What can you say about the sum $x+y$?
Assume not. Then there is an element $x \in F$ which isn't in $G$ and there is a $y \in G$ which isn't in $F$. What can you say about the sum $x+y$?