Proving that $\frac{1}{2}<\frac{2}{3}<\frac{3}{4}<$...$<\frac{n-1}{n}$ In an attempt to find a pattern, I did this:
Let a,b,c,d be non-zero consecutive numbers. Then we have:
$a=a$
$b=a+1$
$c=a+2$
$d=a+3$
This implies:
$\frac{a}{b}=\frac{a}{a+1}$
$\frac{b}{c}=\frac{a+1}{a+2}$
$\frac{c}{d}=\frac{a+2}{a+3}$
I don't know how that helps. I'm greatly seeking your help. Thank you very much.

If $n>1$ is an integer we have
\begin{align*}
n-1&<n\\
\implies \frac{1}{n-1}&>\frac{1}{n}\\
\iff 1-\frac{1}{n-1}&<1-\frac{1}{n}\\
\iff \frac{n-2}{n-1}&<\frac{n-1}{n}
\end{align*}
