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.

Here's an idea. In general you want to show that
$$\frac{a}{a+1} < \frac{a+1}{a+2}.$$
What happens if you multiply $\frac{a}{a+1}$ by $\frac{(a+1)^2}{a+2}$? What do you know about the number $\frac{(a+1)^2}{a+2}$?
