Could you give me some hints, please, to the following problem.
Given $x \in \mathbb{R}$ such that $|x| < 1$. Prove by induction the following inequality for all $n \geq 2$:
$(1-x)^n + (1+x)^n < 2^n$
$1$ Basis:
$n=2$ $(1-x)^2 + (1+x)^2 < 2^2$ $(1-2x+x^2) + (1+2x+x^2) < 2^2$ $2+2x^2 < 2^2$ $2(1+x^2) < 2^2$ $1+x^2 < 2$ $x^2 < 1 \implies |x| < 1$
$2$ Induction Step: $n \rightarrow n+1$ $(1-x)^{n+1} + (1+x)^{n+1} < 2^{n+1}$
(1-x)(1-x)^n + (1+x)(1+x)^n < 2·2^n I tried to split it into $3$ cases: $x=0$ (then it's true), $-1
Could you tell me please, how should I move on. And do I need a binomial theorem here?
Thank you in advance.