Abstract Algebra: Theory and Applications · Problem 2.4.11

Groups — Thomas Judson

Mathematics — Abstract Algebra · induction-bernoulli-inequality

Problem

If xx is a nonnegative real number, then show that (1+x)n−1≥nx(1+x)^n - 1 \geq nx for n=0,1,2,…n = 0, 1, 2, \ldots. Hint. The base case, S(0):(1+x)0−1=0≥0=0⋅xS(0) : (1+x)^0 - 1 = 0 \geq 0 = 0 \cdot x is true. Assume S(k):(1+x)k−1≥kxS(k) : (1+x)^k - 1 \geq kx is true. Then (1+x)k+1−1=(1+x)(1+x)k−1=(1+x)k+x(1+x)k−1≥kx+x(1+x)k≥kx+x=(k+1)x,\begin{aligned} (1+x)^{k+1} - 1 &= (1+x)(1+x)^k - 1 \\ &= (1+x)^k + x(1+x)^k - 1 \\ &\geq kx + x(1+x)^k \\ &\geq kx + x \\ &= (k+1)x, \end{aligned} so S(k+1)S(k+1) is true. Therefore, S(n)S(n) is true for all positive integers nn.

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