Induction proof on natural numbers

Asked Sep 30 '20 at 06:42

Active Sep 30 '20 at 08:21

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I am trying to prove the following using induction, and kind of stuck. I used a computer to validate this sentence. Using $n = 1$ gave the correct result. I assumed that this is true for some $n = k$, but how do I prove for $n = k+1$. Should logarithms be used?

$$2\leq \left(1+\frac{1}{n}\right)^{n}\leq 3$$

I've seen similar solutions, but none with induction, I need a solution using the natural numbers, without limits and calculus.

edited Sep 30 '20 at 08:21

asked Sep 30 '20 at 06:42

user3275222

6

Does this answer your question? Bounding Euler's e by showing $2 \leq (1+ \frac{1}{n})^n < 3$ or https://math.stackexchange.com/questions/2980483/2-1-frac1nn-3-for-all-n-2 or https://math.stackexchange.com/questions/254335/prove-that-5-2-e-3/254339#254339 or Bernoulli's inequality https://en.wikipedia.org/wiki/Bernoulli%27s_inequality – Sumanta Sep 30 '20 at 06:44

Induction proof on natural numbers

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