Odd Convergents of Simple Continued Fraction are Strictly Decreasing

Theorem
Let $n \in \N \cup \{\infty\}$ be an extended natural number.

Let $(a_0, a_1, \ldots)$ be a simple continued fraction in $\R$ of length $n$.

Let $p_0, p_1, p_2, \ldots$ and $q_0, q_1, q_2, \ldots$ be its numerators and denominators.

Let $(C_0, C_1, \ldots)$ be its sequence of convergents.

The odd convergents satisfy $C_1 > C_3 > C_5 > \cdots$

Proof
Let $ k\geq 3$ be an odd integer.

From Difference between Adjacent Convergents But One of Simple Continued Fraction:
 * $C_k - C_{k - 2} = \dfrac {\left({-1}\right)^{k} a_k} {q_k q_{k - 2} } = \dfrac{-a_k}{q_kq_{k-2}}$

By definition of simple continued fraction, $a_k > 0$.

By Convergents of Simple Continued Fraction are Rationals in Canonical Form, $q_k > 0$ and $q_{k-2} > 0$.

Thus $C_k < C_{k-2}$.

Also see

 * Even Convergents of Simple Continued Fraction are Strictly Increasing
 * Properties of Convergents of Continued Fractions