= Solution
{c}
The <Bolzano-Weierstrass theorem> states: \b[every <bounded sequence> of <real numbers> has a <convergent subsequence>.]
By the <monotone subsequence theorem> proved in the preceding part, a <bounded sequence> $(a_n)$ has a <monotone subsequence> $(b_r)$. It inherits the bounds of $(a_n)$. For completeness, the <bounded monotone sequence theorem> follows directly from <completeness of the real numbers>. If $(b_r)$ is increasing, let $L=\sup\{b_r:r\geq1\}$. For every $\varepsilon>0$, the definition of <supremum> supplies $r_0$ with $L-\varepsilon<b_{r_0}\leq L$. Then
$$
r\geq r_0\quad\Longrightarrow\quad L-\varepsilon<b_{r_0}\leq b_r\leq L,
$$
so $b_r\to L$. If $(b_r)$ is decreasing, apply the increasing case to $(-b_r)$, or use its <infimum>. Thus in either case
$$
\boxed{\text{some }n_1<n_2<\cdots\text{ and }L\in\mathbb R\text{ satisfy }a_{n_r}\longrightarrow L.}
$$
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