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I’m searching for an elementary proof that $\bigl(1 + \frac{1}{n}\bigr)^n$ is monotonically increasing for $n = 1, 2, \dots$. The only proofs I can think of use calculus (taking the derivative) or the binomial theorem.

Maybe it is an illusion, but since it seems so obvious from the analogy of bonds that making the compounding interval smaller will increase the the sum of the loan payment, there has to be a rigorous yet elementary argument along these lines?

viuser
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  • What about https://math.stackexchange.com/a/1011442/42969, https://math.stackexchange.com/a/167869/42969, or https://math.stackexchange.com/a/83301/42969 – Martin R Aug 14 '19 at 19:54
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    Why algebra-precalculus? – José Carlos Santos Aug 14 '19 at 20:01
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    With $x_n = (1 +1/n)^n$ you can show that $\frac{x_{n+1}}{x_n} > 1 + \frac{1}{(n+1)^3} > 1$ using only algebra and at one step the Bernoulli inequality $(1 - x)^n > 1 - nx$. While the inequality follows from the binomial theorem, it can also be proved independently by induction. – RRL Aug 14 '19 at 20:37
  • I believe that proof qualifies as "algebra-precalculus". – RRL Aug 14 '19 at 20:39

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