<s>[INST] Proof by induction that $\frac1{n+1} + \frac1{n+2} + \ldots + \frac1{2n} \geq \frac7{12}$ The question is prove that for every integer greater than or equal to 2
$$\frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{2n} \geq \frac{7}{12}$$
So far I have 
Base case let $p(2)$
\begin{align*} 
\frac{1}{2}+ \frac{1}{2+2} + \frac{1}{4} & \geq \frac{7}{12}\\
\frac{5}{6} & \geq \frac{7}{12}
\end{align*}
therefore base case is true.
Let's assume it is a true statement for $p(n+1)$ for all $n$ greater than or equal to $2.$
\begin{align*}
\frac{1}{(n+1)+1} + \frac{1}{(n+1)+2} + \frac{1}{2(n+1)} \geq \frac{7}{12}\\
\frac{1}{n+2} + \frac{1}{n+3} + \frac{1}{2n+2} \geq \frac{7}{12}
\end{align*}
then I wrote that if $n$ is greater than or equal to $2$ is a true statement then the inequality is true. This is a proof by induction and I'm not sure if I concluded properly 
 [/INST] Consider $P(n+1) - P(n)$.  This difference has only three terms, all the middle terms cancel.  And looking at those three terms you can easily show $P(n+1)-P(n)>0$.  That proves that if $P(n)>7/12$ then $P(n+1)$ must also be.
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