<s>[INST] Induction inequality on sum of reciprocals I have to prove that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{1}{2}$ for natural $n$ 
Checking for $n=1$ we have $\displaystyle 1+\frac{1}{2}=\frac{3}{2}\ge \frac{1}{2}$ 
Next we assume that inequality is true for $n$ and for $n+1$ we have: 
$\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}+\frac{1}{2n+1}\ge\frac{1}{2}+\frac{1}{2n+1}\ge\frac{1}{2}$ what is true because $\displaystyle \frac{1}{2n+1} \ge 0$
Is my proof correct ? 
 [/INST] Take a look at the first few cases to get an idea what should happen. The first claim is $\frac11+\frac12\geq\frac12$, the second one is $\frac12+\frac13+\frac14\geq\frac12$, the third one is $\frac13+\frac14+\frac15+\frac16\geq\frac12$.
So to get from $\frac1n+\dots+\frac1{2n}$ to $\frac1{n+1}+\dots+\frac1{2(n+1)}$ you need to subtract $\frac1n$ and add $\frac1{2n+1}+\frac1{2n+2}$. Thus the difference of the adjacent sums is $\frac1{2n+1}+\frac1{2n+2}-\frac1n=-\frac{3n+2}{n(2n+1)(2n+2)}$. This is negative, so the sum keeps getting smaller as $n$ grows. Therefore such an induction cannot work.
A proof without induction is easier: Your sum has $n+1$ terms and the smallest one is $\frac1{2n}$. Thus the sum is at least $(n+1)\times\frac1{2n}=\frac{n+1}{2n}>\frac12$.
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