Denominators of Simple Continued Fraction are Strictly Positive

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Theorem

Let $n \in \N \cup \set \infty$ be an extended natural number.

Let $\tuple {a_0, a_1, \ldots}$ be a simple continued fraction in $\R$ of length $n$.

Let $q_0, q_1, q_2, \ldots$ be its denominators.


Then for $0 \leq k \leq n$ we have $q_k > 0$.


Proof

Follows from:

$q_0 = 1$ by definition
Denominators of Simple Continued Fraction are Strictly Increasing.

$\blacksquare$