let be the open set. We have to show that is an open set of the metric space (endowed with the metric which is the metric induced by the norm . Note that the statement of the problem says is a distance but it is wrong, such defined is a norm. We used this norm to induce the distance function (or metric) ).
This means that for every we must find an open ball in the space with center at such that the whole ball is a subset of .
How do open balls of look like? Given some an open ball of with center at some point and radius is the set . So they're open intervals.
Since implies and is an open set in , there must exist an open ball of with center at and some radius such that this whole ball is a subset of . This open ball is the set
Now if we intersect this open ball with , we get exactly an open ball of the space centered at :
This ball is obviously a subset of (because the original ball is a subset of ).
We conclude that the set is an open set of the space .