Bolzano-Weierstrass Theorem/Proof 1

Theorem

Every bounded sequence of real numbers has a convergent subsequence.

Proof

Let $\sequence {x_n}$ be a bounded sequence in $\R$.

By the Peak Point Lemma, $\sequence {x_n}$ has a monotone subsequence $\sequence {x_{n_r} }$.

Since $\sequence {x_n}$ is bounded, so is $\sequence {x_{n_r} }$.

Hence, by the Monotone Convergence Theorem (Real Analysis), the result follows.

$\blacksquare$

Also see

• Heine-Borel Theorem

Source of Name

This entry was named for Bernhard Bolzano and Karl Weierstrass.

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Bolzano-Weierstrass Theorem: $\S 5.10$

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces