<s>[INST] Prove $0.9999^{\!101}<0.99<0.9999^{\!100}$ Prove
$$0.9999^{\!101}<0.99<0.9999^{\!100}$$
I think its original idea is 
$$(1-x)^{(1-\frac{1}{x})}<(1+x)^{(\frac{1}{x})} \tag{I can't prove!}$$
For $x=100^{\!-1}\,\therefore\,(1-x)^{(1-\frac{1}{x})}<(1+x)^{(\frac{1}{x})}\,\therefore\,0.99^{\!-99}<1.01^{\!100}$.
Furthermore $0.99^{\!-99}\times0.99^{\!100}=0.99<1.01^{\!100}\times0.99^{\!100}=0.9999^{\!100}$.
For $x=-100^{\!-1}\,\therefore\,(1-x)^{(1-\frac{1}{x})}<(1+x)^{(\frac{1}{x})}\,\therefore\,1.01^{\!101}<0.99^{\!-100}$.
Furthermore $1.01^{\!101}\times0.99^{\!101}=0.9999^{\!101}<0.99^{\!-100}\times0.99^{\!101}=0.99$.
 [/INST] $$0.99<0.9999^{100}$$ it's
$$1-\frac{1}{100}<\left(1-\frac{1}{100^2}\right)^{100},$$ which is true by Bernoulli:
$$\left(1-\frac{1}{100^2}\right)^{100}>1-100\cdot\frac{1}{100^2}=1-\frac{1}{100}.$$
$$0.9999^{101}<0.99$$ it's 
$$\left(1-\frac{1}{100^2}\right)^{101}<1-\frac{1}{100}$$ or
$$\left(1-\frac{1}{100^2}\right)^{100}\left(1+\frac{1}{100}\right)<1$$ or
$$\left(1-\frac{1}{100^2}\right)^{100}<1-\frac{1}{101},$$ which is true because
$$\left(1-\frac{1}{100^2}\right)^{100}<1-\binom{100}{1}\cdot\frac{1}{100^2}+\binom{100}{2}\cdot\frac{1}{100^4}<1-\frac{1}{101}.$$
I used the following lemma.

Let $0<x<1$ and $n>2.$ Prove that:
  $$(1-x)^n<1-nx+\frac{n(n-1)}{2}x^2.$$

Proof.
We need to prove that $f(x)>0,$ where
$$f(x)=1-nx+\frac{n(n-1)}{2}x^2-(1-x)^n.$$
Indeed, $$f'(x)=-n+n(n-1)x+n(1-x)^{n-1}$$ and
$$f''(x)=n(n-1)-n(n-1)(1-x)^{n-2}=n(n-1)\left(1-(1-x)^{n-2}\right)>0.$$
Thus, $$f'(x)>f'(0)=0,$$
$$f(x)>f(0)=0$$ and the lemma is proven. 
Now, take $x=\frac{1}{100^2}$ and $n=100.$
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