Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. For $n=2$, we have $LHS=1+2x$, and $RHS=(1+x)^2=1+2x+x^2$, thus $LHS\le RHS$ and $P(2)$ is true.
3. Assume $P(k), k\gt2$ is true, that is $1+kx\le(1+x)^{k}$
4. For $n=k+1$, we have $RHS=(1+x)^{k+1}=(1+x)(1+x)^{k}\ge(1+x)(1+kx)=1+kx+x+kx^2=1+(k+1)x+kx^2\ge1+(k+1)x=LHS$
5. Thus $P(k+1)$ is also true and we have proved the statement by mathematical induction.