Tag: alternative-proof

real analysis – Proofs of AM-GM inequality

The Question : 108 people think this question is useful The arithmetic – geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I’m looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone knew a proof that is

number theory – $n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn’t use Chebyshev’s theorem?

The Question : 110 people think this question is useful If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev’s theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and $2n-2$. A proof can be found here. Two weeks and four