The union of two subspaces is not alwys a subspace

Question

$F$ and $G$ are two subspaces of $E$ over $K$, how can I show that, $F \cup G$ is a subspace of E over K $\Leftrightarrow$ $F \subseteq G$ or $G \subseteq F$ ? $\Leftarrow$ is obvious, but what about $\Rightarrow$ ?

Answer 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 $x+y$?