# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

2,281 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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### 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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### Prove that $n < 2^{n}$ for all natural numbers $n$. [duplicate]

Prove that $n < 2^{n}$ for all natural numbers $n$. I tried this with induction: Inequality clearly holds when $n=1$. Supposing that when $n=k$, $k<2^{k}$. Considering $k+1 <2^{k}+1$, ...
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### Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$

I met an inequality, I ask, do not mathematical induction to prove that: Prove $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$ without using induction
### How can I prove that $xy\leq x^2+y^2$? [closed]
How can I prove that $xy\leq x^2+y^2$ for all $x,y\in\mathbb{R}$ ?