= Bolzano-Weierstrass theorem
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{title2=$x_{n_j}\to\ell$}
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Every <bounded sequence> of <real numbers> contains a <subsequence> with a finite <limit of a sequence>. If $|x_n|\leq M$ for every <positive integer> $n$, there are <integers> $n_1<n_2<\cdots$ and a <real number> $\ell$ such that
$$
\lim_{j\to\infty}x_{n_j}=\ell.
$$
The increasing indices matter: a <subsequence> retains the original order and selects terms arbitrarily far along the original <sequence>. The theorem guarantees the existence of at least one such selection; it does not claim that the entire <sequence> converges or that the <subsequential limit> is unique.<Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl/2.3 Limit superior, limit inferior, and Bolzano–Weierstrass>[Basic Analysis, Volume I by Jiří Lebl, §2.3.3, Theorem 2.3.8: every bounded real sequence has a convergent subsequence]{ref}
For example, the <bounded sequence> $x_n=(-1)^n+1/(n+1)$ has two distinct <subsequential limits>:
$$
x_{2j}=1+\frac1{2j+1}\longrightarrow1,\qquad
x_{2j-1}=-1+\frac1{2j}\longrightarrow-1.
$$
The even terms form a <convergent sequence>, and so do the odd terms. The full <sequence> cannot have a <limit of a sequence> because its two <subsequences> have different limits. Its <limit superior> is $1$ and its <limit inferior> is $-1$.
\Image[/bounded-alternating-sequence-two-subsequential-limits.png]
{title=A <bounded sequence> can have two different <subsequential limits>}
{description=Terms of $x_n=(-1)^n+1/(n+1)$ are colored by even or odd index. The dashed lines mark $1$ and $-1$. Each color approaches its own limit, while the entire <sequence> keeps moving between the two groups. The values are exact model calculations, not observations.}
This is useful when a construction produces many candidate points but does not immediately produce a <convergent sequence>. For a <continuous function> on a nonempty closed bounded <feasible region> in a finite-dimensional <Euclidean space>, take a <sequence> whose objective values approach the <infimum>. A <convergent subsequence> supplies a candidate point; a <closed subset> retains that point, and <continuity> passes the objective values to the <limit of a sequence>. This is the mechanism behind attainment in the <extreme value theorem>. The <diagonal subsequence argument> extends the real-sequence theorem to finite-dimensional <Euclidean space>.
The theorem is named for <Bernard Bolzano> and <Karl Weierstrass>.<Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl/2.3 Limit superior, limit inferior, and Bolzano–Weierstrass>[Basic Analysis, Volume I by Jiří Lebl, opening footnote to §2.3: the theorem's two namesakes]{ref}
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