The Mathematical Proof Thread

[Succubus]

Prove Bernoulli’s Identity: For every real number x > -1 and every positive integer n, (1 + x)^n ≥  1 + nx

Proof

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Let y > -1 be an arbitrary real number.  We will demonstrate that for every positive integer n, (1 + y)^n ≥ 1 + ny.  We will establish this by induction.  To that end, let n=1.  Then (1 + y)^1 = 1 + y ≥ 1 + (1)y = 1 + y, which is true.  Assume that (1 + y)^k ≥ 1 +ky for some positive integer k.  We must show that (1 + y)^(k+1) ≥ 1 + (k +1)y. 

Now, (1 + y)^(k+1) = [(1 + y)^k]*(1 + y) ≥ (1 + ky)*(1 + y) (1) = 1 + y +ky + ky^2 ≥ 1 + y + ky (2) = 1 + (k + 1)y (please note that we made use of the induction hypothesis to reach result (1), and we made use of the fact that k is a positive integer and y^2 is non-negative to obtain result (2)).  So, by the principle of mathematical induction, it follows that for every positive integer n, (1 + y)^n ≥ 1 + ny. 

Hence, since y > -1 is an arbitrary real number, then it follows that for every real number x > -1 and every positive integer n, (1 + x)^n ≥ 1 + nx.  The proof is complete.

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Looks sound enough to me.