Number Theory (Prove by Mathematical Induction)

Question

Prove by Mathematical Induction:

If x ≥ 0, then ∀n ∈N, (1+x)ⁿ ≥1 + xⁿ

Answer 1

For n=1 it is obvious. Assume now that the claim is true for some n. Then (1+x)^{n+1}=(1+x)^{n}(1+x) which is greater or equal to (by the induction hypothesis) (1+x^{n})(1+x)=1+x+x^{n}+x^{n+1} which is greater or equal to just 1+x^{n+1} since x is a nonnegative variable. By induction the proof is complete.