Proving $\frac{n}{n+1} < \frac{n+1}{n+2}$ by induction? I have the inequality
$\frac{n}{n+1} < \frac{n+1}{n+2}$
I'm not sure how to go about proving it. I've started by testing with n = 1, which results in
$\frac{1}{2} < \frac{2}{3}$ which is true
I then assume true for n = k and have to prove that it is true for n = k + 1, but I don't know how to start manipulating
$\frac{k}{k+1} < \frac{k+1}{k+2}$
to become
$\frac{(k+1)}{(k+1)+1} < \frac{(k+1)+1}{(k+1)+2}$
How do I go about doing this?

if you need induction:
\begin{gather*}
n(n+2)<(n+1)^2 \\
\begin{aligned}
(n+1)(n+3)={}&n(n+2+1)+(n+3)=(n(n+2)+n+n+3)<(n+1)^2+2n+3={} \\
{}={}&(n^2+4n+4)=(n+2)^2.
\end{aligned}
\end{gather*}
