<s>[INST] Proof an Inequality of a sequence $\frac{1}{2^1-1}+\frac{1}{2^2-1}+\cdots+\frac{1}{2^n-1}\lt \frac{5}{3}$ I have tried to amplify it like this 
 $$\frac{1}{2^1-1}+\frac{1}{2^2-1}+\cdots+\frac{1}{2^n-1}\lt 1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^{n-1}}=2-\frac{1}{2^{n-1}} $$
which obviously too big. Any proper ways to amplify it ?
 [/INST] Maybe you are overdoing it. Consider leaving a first few terms untouched and then "amplify" the rest like you did. 
$$S < \dfrac{1}{2-1}+\dfrac{1}{2^2-1}+\dfrac{1}{2^3}+...+\dfrac{1}{2^n}=...?$$
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