Prove this is a subspace Let $ W_1, W_2$ be subspace of a Vector Space $V$.
Denote $W_1+W_2$ to be the following set
$$W_1+W_2=\left\{u+v, u\in W_1, v\in W_2\right\}$$
Prove that this is a subspace.
I can prove that the set is non emprty (i.e that it houses the zero vector).
pf:  Since $W_1 , W_2$ are subspaces, then the zero vector is in both of them.
$$\mathbb{O}_V+\mathbb{O}_V=\mathbb{O}_V$$
but I can't wrap my head around the closure of addition and scalar multiplication.

Denote $ W = W_1 + W_2 $ and let $v \in W_1$, $w \in W_2$, $W \ni x = v + w $. We need to check that $cx \in W$. But observe $cx = c v + c w$. Since $W_i$ are vector spaces $cv$ and $cw$ are in them. Do something similar for addition (regroup terms, use closure under addition in the subspaces.
