Answer
See below.
Work Step by Step
Proofs using mathematical induction consists of two steps:
1) The base case: here we prove that the statement holds for the first natural number.
2) The inductive step: assume the statement is true for some arbitrary natural number $n$ larger than the first natural number, then we prove that then the statement also holds for $n + 1$.
Hence here:
1) For $n=1$: if $0\lt x\lt1$, then $0\lt x^1=x\lt1$.
2) Assume for $n=k: 0\lt x^k\lt1$. Then for $n=k+1$:
$x^{k+1}=x^k\cdot x$ and $0\lt x^k\lt1$ and $0\lt x\lt1$, thus their product will also be between $0$ and $1$.
Thus we proved what we wanted to.