The proof of the triangle inequality of this norm Let $V$ be a normed vector space and $W$ a closed subspace of $V$. It is possible to define a norm on $V/W$ (the quotient space) by defining $\|v + W \|_{V/W} = \inf_{w \in W} \|v + w\|$. As an exercise I tried to prove the triangle inequality:
$$ \|v + v' + W\|_{V/W} \le \|v + W \|_{V/W} + \|v' + W \|_{V/W}$$
for all $v,v' \in V$.
Would anyone please check my proof?
Proof: 
Observe that $2 \cdot W = W$ and that for all $v,v' \in V$ and all $w \in W$ the following inequality holds:
$$ \|v + v' + 2w\| \le \|v+w\| + \|v' + w\|$$
Then in particular for all $v,v' \in V$ and all $w \in W$:
$$ \inf_{w \in W} \|v + v' + 2w\| = \inf_{w \in 2W} \|v + v' + w\| = \|v + W \|_{V/W} \le \|v+w\| + \|v' + w\|$$
Because this inequality holds for all $v,v' \in V$ and all $w \in W$ it follows that
$$ \|v + W \|_{V/W} \le \|v+w\| + \inf_{w \in W}  \|v' + w\| $$
for all $v \in V$ and all $w \in W$ and because the inequality holds for all $v \in V$ and all $w \in W$ it follows that
$$ \|v + W \|_{V/W} \le \inf_{w \in W} \|v+w\| + \inf_{w \in W}  \|v' + w\| = \|v + W \|_{V/W} + \|v' + W \|_{V/W}$$
where the last step is really two steps at once. 
 A: Unfortunately, not totally correct. You can not take infimum of two terms depending on the same variable and replace it with the sum of the infima:
$$
\inf_{a\in A} \big(f(a)+g(a)\big) \ge \inf_{a\in A} f(a)+ \inf_{a\in A} g(a),
$$
whereas
$$
\inf_{a\in A} f(a)+ \inf_{a\in A} g(a)=\inf_{a,a'\in A} \big(f(a)+g(a')\big).
$$
I would fix it as follows:
Proof. For all $v,v' \in V$ and all $w_1,w_2 \in W$ the following inequality holds:
$$ \|v + v' + w_1+w_2\| \le \|v+w_1\| + \|v' + w_2\|$$
Then in particular for all $v,v' \in V$ and all $w_1,w_2 \in W$:
$$ \inf_{w_1,w_2 \in W} \|v + v' + w_1+w_2\| = \inf_{w \in W} \|v + v' + w\| = \|v +v'+ W \|_{V/W} \le \|v+w_1\| + \|v' + w_2\|.$$
Thus, for every $w_1,w_2$ the quantity $\|v+w_1\| + \|v' + w_2\|$ is greater or equal with $\|v + W \|_{V/W}$, and so is the infimum this quantity over all $w_2\in W$, i.e.,
$$
\|v + W \|_{V/W} \le \|v+w_1\| +\inf_{w_2\in W}\|v'+w_2\|=\|v+w_1\| +\|v' + W \|_{V/W}.
$$
Using the same argument,  $\|v+w_1\| +\|v' + W \|_{V/W}$ is greater or equal to $\|v + W \|_{V/W}$, for all $w_1\in W$, and so is the infimum this quantity over all $w_1\in W$, i.e.,
$$
\|v + W \|_{V/W} \le \inf_{w_1\in W}\|v+w_1\| +\|v' + W \|_{V/W}=\|v + W \|_{V/W}+\|v' + W \|_{V/W}
$$
QED.
