How do you prove this by induction?

Question

Prove Bernoulli’s inequality by induction: For an arbitrary x ∈ R, such that x ≥ 1, prove by induction on n that

(1 +x)n ≥ 1+nx

for any positive integer n.

Answer 1

Since (1 + x)^1 = 1 + x = 1 + 1x, the statement is true when n = 1. Now suppose n > 1 and the assertion is true for n. Then (1 + x)ⁿ ≥ 1 + nx, so

(1 + x)ⁿ⁺¹ = (1 + x)ⁿ(1 + x) ≥ (1 + nx)(1 + x) = 1 + nx + x + nx² = 1 + (n+1)x + nx² ≥ 1 + (n+1)x

and the assertion holds for n + 1. By induction on n, the statement holds for all n ≥ 1.