Prove by Induction that: $1+nx\le (1+x)^n$?

Question

$1+nx\le (1+x)^n$, for all real numbers $x>-1$ and integers $n\ge 2$.

Can you please also explain a little of the basic step and the inductions step.

Thank you in advance.

Answer 1

Notice the case $n=2$ is the case

$$ 1 + 2x \leq (1 + x)^2 $$

which is true(as you should verify). Suppose result holds for $n$, we show it holds for $n+1$,

$$ (1+x)^{n+1} = (1+x)^n (1+x) \geq (1 + nx)(1+x) = 1 + x + nx + n x^2 = 1 +x(n+1) + nx^2 \geq 1 + x(n+1)$$

since $x^2 \geq 0 $ for all $x$.