Suppose $0 < a < b$. Which of the following continued
  fractions is bigger and why? <br />
  <br />
  \[ \frac{1}{2+\frac{1}{3+\frac{1}{a}}} \] \par or \[
  \frac{1}{2+\frac{1}{3+\frac{1}{b}}} \] <br />
  <br />
  Suppose the fractions are continued in the same way, then which
  is the bigger in the following pair and why? <br />
  <br />
  \[ \frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{a}}}} \] <br />
  <br />
  or the same thing with b in place of a. <br />
  <br />
  Now compare: $${1\over\displaystyle 2 + { 1 \over \displaystyle
  3+ { 1\over \displaystyle 4 + \dots + {1\over\displaystyle 99+
  {1\over \displaystyle {100 + {1 \over \displaystyle a}} }}}}}$$
  <br />
  <br />
  and the same thing with $b$ in place of $a$.<br />