Relative Size of Convergents of Finite Continued Fractions

Theorem (Relative Size of Convergents)
For any finite continued fraction
$[ a_1,a_2,...,a_{n-1},a_n ]$
, the odd convergents
$C_{2r-1}$
form a strictly increasing sequence
$C_1 \lt C_3 \lt C_5 ...$
and the even convergents
$C_{2r}$
form a strictly decreasing sequence
$C_2 \gt C_4 \gt C_6 ...$
/Br /> Also every even convergent is greater than every odd convergent.
Proof
From the Properties of Convergents of Finite Continued Fractions,
$a_k \gt 0$
for for
$k \ge 2$
and
$q_k \gt 0$
for
$k \ge 1$
, so for
$k \ge 3$
,
$C_k-C_{k-2}=\frac{p_k}{q_k}-\frac{p_{k-2}}{q_{k-2}}=\frac{(-1)^{k-1}a_k}{q_kq_{k-1}}$
has the same sign as
$(-1)^{k-1}$
so is positive for odd
$k$
and negative for even
$k$
, showing that the odd convergents strictly increase and the even convergents strictly decrease.
$C_k-C_{k-1}=\frac{p_k}{q_k}-\frac{p_{k-1}}{q_{k-1}}=\frac{(-1)^{k}}{q_kq_{k-1}}$

Hence the even convergent is larger than the succeeding odd convergent.