A reduced rational number $p/q$, with $p \in \mathbb{Z}$ and $q \in \mathbb{N}$, is called a strict best approximation of the second kind to $\theta$ if
for every $(r,s) \in \mathbb{Z} \times \mathbb{N}$ satisfying $1 \le s \le q$ and $r/s \ne p/q$.
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Then the strict best approximations of the second kind to $\theta$ are precisely the convergents $p_n/q_n$ with $n \ge 1$, together with $p_0/q_0$ exactly in the case $a_1 \ge 2$.
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If instead one defines a weak best approximation of the second kind under the full-range convention by requiring
for every $(r,s) \in \mathbb{Z} \times \mathbb{N}$ satisfying $1 \le s \le q$ and $r/s \ne p/q$, then the weak best approximations of the second kind are the same rational numbers as the strict best approximations of the second kind. Thus, under this convention, they are precisely the convergents $p_n/q_n$ with $n \ge 1$, together with $p_0/q_0$ exactly when $a_1 \ge 2$.
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In particular, for $n \ge 0$ and $1 \le m < a_{n+1}$, the intermediate continued-fraction candidates
are not additional second-kind best approximants under this convention. They are the additional candidates relevant to first-kind best approximation, where the comparison is made using $|\theta - p/q|$ rather than $|q\theta - p|$.