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By using the definition $$\pmatrix{p_n&p_{n-1}\\q_n&q_{n-1}} = \pmatrix{a_0&1\\1&0} \pmatrix{a_1&1\\1&0} \cdots \pmatrix{a_n&1\\1&0}$$ I need to show that $p_n/q_n$ is the continued fraction $[a_0;a_1, ..., a_n]$

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The continued fraction notation means $$[a;m] = a + 1/[m].$$

From $$\pmatrix{p_0&p_{-1}\\q_0&q_{-1}} = \pmatrix{a_N&1\\1&0}$$ we take the base case to be $(p_0,q_0) = (a_N,1)$, $(p_{-1},q_{-1}) = (1,0)$. This holds trivially.

We show the rest by induction. If $p_n/q_n$ is the convergent for $[m]$ then $$p_{n+1}/q_{n+1} = a + 1/[m] = a + q_n/p_n = \frac{a p_n + q_n}{p_n}.$$

So to complete the proof we check $$\pmatrix{p_{n+1}&p_{n}\\q_{n+1}&q_{n}} = \pmatrix{a p_{n} + q_n &p_{n}\\p_{n}&q_{n}} = \pmatrix{a&1\\1&0}\pmatrix{p_n&p_{n-1}\\q_n&q_{n-1}}.$$

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