Suppose that $0< a< b$. Which of the following continued fractions is bigger? Why? $\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{a}}} \quad \textrm{or} \quad \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{b}}}$
Suppose the fractions are continued in the same way, then which is bigger in the following pair and why? $\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{a}}}} \quad \textrm{or} \quad \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{b}}}}$
Now compare: $\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\dotsb+\cfrac{1}{99+\cfrac{1}{100+\cfrac{1}{a}}}}}}$ and the same thing with $b$ in place of $a$.