Exercise 499Logic, sets and proofs › Proof techniques › InductionL1Added 3 Nov 2026

Bernoulli's inequality by induction

#induction#Bernoulli#inequality

Statement

Prove by induction: for all $x\ge-1$ and all $n\ge1$ , $(1+x)^n\ge1+nx$ .

Hint

Base case $n=1$ ; inductive step: multiply by $1+x\ge0$ .

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Base case. $n=1$ : $(1+x)^1=1+x$ . ✓

Inductive step. Assume $(1+x)^n\ge1+nx$ . Since $x\ge-1$ , $1+x\ge0$ , so
$(1+x)^{n+1}\ge(1+nx)(1+x)=1+(n+1)x+nx^2\ge1+(n+1)x.$

Conclusion. The inequality holds.