Let
p
n
/
q
n
{p_n}/{q_n}
and
A
n
/
B
n
{A_n}/{B_n}
denote the convergents of, respectively, the regular and the nearest integer continued fraction expansion of the irrational number x. There exists a function
k
(
n
)
k(n)
such that
A
n
/
B
n
=
p
k
(
n
)
/
q
k
(
n
)
{A_n}/{B_n} = {p_{k(n)}}/{q_{k(n)}}
. Adams proved that for almost all x one has
lim
k
(
n
)
/
n
=
log
2
/
log
G
\lim k(n)/n = \log 2/\log G
,
G
=
1
2
(
1
+
5
)
G = \frac {1}{2}(1 + \sqrt 5 )
. Here we present a shorter proof of this result, based on a simple expression for
k
(
n
)
k(n)
and the ergodicity of the shift operator, connected with the nearest integer continued fraction.