Prove what fraction of a hypercube in m dimensions is within the extreme 10% of it.

Question

I am attempting to find what proportion of the points in a hypercube are in the boundary region, i.e. have a $j$ for which $0\le x_j\le0.05$ or $0.95\le x_j\le1$, i.e. have at least one dimension within the most extreme 10% of possible values. By inspection I have found that this is represented by $$0.1\sum_{n=0}^{m-1}0.9^n$$ Where $m$ is the number of dimensions of the hypercube. By induction (or otherwise, I just feel the proof might be fairly straight forward by induction), how would I prove this result?

Answer 1

It feels so easy when the complement is considered: what proportion of points have none of their coordinates in the extreme 10% of values? $0.9^m$. Thus the proportion of points in the boundary region (at least one coordinate is extreme) is $1-0.9^m$.

Notwithstanding this, the given summation reduces to $1-0.9^m$ and is thus correct.