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, $x + y \in W_1 + W_2$?

Seems to me that there is a fundamental misunderstanding of quantification here. How do I show that something is in $W_1+W_2$? All that is necessary is to find an element of $W_1$ and an element of $W_2$ that add up to the given thingie. Starting out with $w_1$ in $W_1$, the “an” elt of $W_1$ is it itself, and the “an” elt of $W_2$ is the zero elt. 
