Bolzano–Weierstrass Theorem

Every bounded real sequence has a convergent subsequence.

The idea

Theorem (Bolzano–Weierstrass).

Every bounded sequence of real numbers has a convergent subsequence.

The figure shows the terms of a bounded sequence in $[-M, M]$ and a point $x$ that a subsequence of them converges to.

A subsequence keeps infinitely many terms of $a_{1}, a_{2}, a_{3}, \dots$ in their original order and discards the rest — the even-index terms $a_{2}, a_{4}, a_{6}, \dots$, say. We write it $a_{n_{1}}, a_{n_{2}}, \dots$ with $n_{1} < n_{2} < \cdots$.

The only hypothesis is boundedness: every term lies in some interval $[-M, M]$. The theorem does not claim that the sequence itself converges — a bounded sequence may oscillate forever. It claims that some infinite selection of its terms converges; the existence of the subsequence is the content.

Boundedness also cannot be dropped: a sequence whose terms run off to infinity carries every subsequence with it.

Ways to work on it

• Walkthrough. Every bounded sequence has a convergent subsequence — and why boundedness is essential.
• Practice. Find the limit of a named convergent subsequence of a bounded sequence.
• Hardest. Extract a convergent subsequence and count the cluster points of a bounded sequence.

Not sure where to start? Take the ten-question placement test.