Showing that $W_1\subseteq W_1+W_2$ I found this question and answer on UCLA's website:
Let $W_1$ and $W_2$ be subspaces of a vector space $V$ .
Prove that $W_1 +W_2$ is a subspace of $V$ that contains both $W_1$ and $W_2$.
The answer given:

First, we want to show that $W_1 \subseteq W_1 +W_2$. Choose $x \in W_1$. Since
  $W_2$ is a subspace, $0 \in W_2$ where $0$ is the zero vector of $V$ . But $x = x + 0$ and $x \in W_1$. Thus, $x \in W_1 + W_2$ by definition. Ergo, $W_1 \subseteq W_1 + W_2$. We also must show that $W_2 \in W_1 + W_2$, but
  this result is completely analogous (see if you can formalize it).

My question:
Why is it enough to show that $x + 0 \in W_1 + W_2$, $0$ is just one element in $W_2$, why don't we have to show, for example, $x + y \in W_1 + W_2$?

$W_1+W_2=\{w_1+w_2: w_1\in W_1,w_2\in W_2\}$. To show that an element belongs to this set, we just need to show that it can be written in the form $w_1+w_2$ for some $w_1\in W_1$ and some $w_2\in W_2$.
