I want to show that $$\forall\space n\ge 1: \left(1+\frac{1}{2n}\right)^n\le 2$$ using Bernouilli's inequality. I don't see a starting point since the inequality $$(1+x)^n\ge 1+nx$$ $\forall\space n\ge 1$ and $x\ge -1$ only gives us a "greater or equal" starting point. Any tips?
2026-03-25 14:19:18.1774448358
Bernouilli inequality for exponential sequence
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Hint:
$$\left(1+\frac{1}{2n}\right)^n \le \frac{1}{\left(1-\cfrac{1}{2n}\right)^n} \le \frac{1}{1 - n \cdot \cfrac{1}{2n}}$$