# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

2,317 questions
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### How to prove that $\log(x)<x$ when $x>1$?

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
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### Proving : $\bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n}$

How could we prove that this inequality holds $$\left(1+\frac{1}{n+1}\right)^{n+1} \gt \left(1+\frac{1}{n} \right)^{n}$$ where $n \in \mathbb{N}$, I think we could use the AM-GM inequality for ...
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### How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$? [duplicate]

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
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### Proof that $n^2 < 2^n$

How do I prove the following statement by induction? $$n^2 \lt 2^n$$ $P(n)$ is the statement $n^2 \lt 2^n$ Claim: For all $n \gt k$, where $k$ is any integer, $P(n)$ (since $k$ is any integer, I ...