Explain briefly how a positive irrational number $x$ gives rise to a continued fraction

$$a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\ldots}}}$$

with the $a_{j}$ non-negative integers and $a_{j} \geqslant 1$ for $j \geqslant 1$.

Show that, if we write

$$\left(\begin{array}{ll} p_{n} & p_{n-1} \\ q_{n} & q_{n-1} \end{array}\right)=\left(\begin{array}{cc} a_{0} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{1} & 1 \\ 1 & 0 \end{array}\right) \cdots\left(\begin{array}{cc} a_{n-1} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{n} & 1 \\ 1 & 0 \end{array}\right)$$

then

$$\frac{p_{n}}{q_{n}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{n-1}+\frac{1}{a_{n}}}}}$$

for $n \geqslant 0$.

Use the observation [which need not be proved] that $x$ lies between $p_{n} / q_{n}$ and $p_{n+1} / q_{n+1}$ to show that

$$\left|p_{n} / q_{n}-x\right| \leqslant 1 / q_{n} q_{n+1}$$

Show that $q_{n} \geqslant F_{n}$ where $F_{n}$ is the $n$th Fibonacci number (thus $F_{0}=F_{1}=1$, $\left.F_{n+2}=F_{n+1}+F_{n}\right)$, and conclude that

$$\left|\frac{p_{n}}{q_{n}}-x\right| \leqslant \frac{1}{F_{n} F_{n+1}}$$