$$1- \frac{ 1 }{ 1- \frac{ 1 }{ 1- \frac{ 1 }{ 9 } } }$$
$9$
$$1-\frac{1}{1-\frac{1}{\frac{9}{9}-\frac{1}{9}}}$$
$$1-\frac{1}{1-\frac{1}{\frac{9-1}{9}}}$$
$$1-\frac{1}{1-\frac{1}{\frac{8}{9}}}$$
$$1-\frac{1}{1-1\times \frac{9}{8}}$$
$$1-\frac{1}{1-\frac{9}{8}}$$
$$1-\frac{1}{\frac{8}{8}-\frac{9}{8}}$$
$$1-\frac{1}{\frac{8-9}{8}}$$
$$1-\frac{1}{-\frac{1}{8}}$$
$$1-1\left(-8\right)$$
$$1-\left(-8\right)$$
$$1+8$$
$$9$$
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$3^{2}$
