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Binomial inequality

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The binomial inequality is a statement about raising a sum to a natural power:

\forall x \in \R, x \ge -1, and \forall n \in \N : (1 + x)^n \ge 1 + nx.

It is essentially a truncated version of the binomial theorem, that is easily proved by induction.

Proof Edit

Check it's true for n = 1 : (1 + x) \ge 1 + x, clearly.

Now assume true for some n = k \in \N : (1 + x)^k \ge 1 + kx.

As  1 + x \ge 0, we can multiply that inequality through by (1 + x) and preserve the inequality:

 (1 + x)^{k+1} \ge (1 + x)(1 + kx) = 1 + x + kx + kx^2.

As kx^2 \ge 0 \forall x \in \R, \forall n \in \N : 1 + x + kx + kx^2 \ge 1 + (1 + k)x.

So, (1 + x)^{k+1} \ge 1 + (k + 1)x, so by induction it is true for all n \in \N.

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