About quotient space problem

Question

Quotient space: Let V be any vector space and $W \subset V$ a subspace. For any $v\in V$, let $v+W$ and denote the set: $v+W=\{v+w\mid w \in W\} \subset V $, called the coset of W containing v. Let $V/W=\{v+W\mid v \in V\}$ which we call the quotient space of $V$ modulo $W$.

Problem: prove that $v_1+W=v_2+W$ if and only if $v_1-v_2 \in W$.

$\rightarrow$: I am thinking W is a subspace, to prove that $v_1-v_2 \in W$, define $v_1,v_2 \in V$, can I say that $v_1+W-(v_2+W)= v_1+w-v_2-w=v_1-v_2$. Now I am hitting a dead end.

Answer 1

If $v_1+W=v_2+W$, then, since $v_1\in v_1+W$, then $v_1\in v_2+W$ and therefore $v_1-v_2\in W$.

And if $v_1-v_2\in W$, take $v\in v_1+W$. Then $v-v_1\in W$ and$$v=\overbrace{(v-v_1)}^{\phantom W\in W}-\overbrace{(v_1-v_2)}^{\phantom W\in W}\in W$$and, by a similar argument, if $v\in v_2+W$, then $v\in v_1+W$.

Answer 2

Note that $v_1\in v_1+W$ and $v_1\in v_2+W$ means that there exists $w\in W$ with $v_1=v_2+w$.