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I know how to do the proof for $U+W$ being a subspace of something, say, $W$. But if I wanted to do $U-W$ being a subspace would I do $v_1=u_1+w_1$ and $v_2=u_2+w_2$ and do $v_1-v_2$ or would I do $v_1=u_1-w_1$ and $v_2=u_2-w_2$ and the do $v_1-v_2$

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    Note that, if $V$ is a subspace, then $-v\in V \iff v\in V$. So $-V=V$. – 2017-02-14
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    What do you mean by $U-W$ ? If you mean $U-W=\{u-w : u \in U, w \in W \}$, then $U-W = U+W$, since $W$ is a subspace. – 2017-02-14
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    I'm afraid that, in your sense $U-W$ is nothing else but $U+W$. – 2017-02-14
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    Sorry I mean if V is a subspace and U and W are subspaces of V then U-W is a subspace also. – 2017-02-14

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If you meant $\;U-W=U+W\;$ then it is the same you already proved, but if you meant the set theoretical difference $\;U\setminus W\;$ , then observe that

$$\;0\in U,\,0\in W\implies 0\notin U\setminus W\implies U\setminus W\;\;\text{ is not a subspace}$$