Bolzano-Weierstrass Theorem/Lemma 2

Theorem

Let $S$ be a non-empty subset of the real numbers such that its infimum $\map \inf s$ exists.

Let $\map \inf s \notin S$.

Then $\map \inf s$ is a limit point of $S$.


Proof

The proof follows exactly the same lines as Lemma $1$.

$\blacksquare$


Also known as

Some sources refer to the Bolzano-Weierstrass Theorem as the Weierstrass-Bolzano Theorem.

It is also known as Weierstrass's Theorem, but that name is also applied to a completely different result.


Source of Name

This entry was named for Bernhard Bolzano and Karl Weierstrass.


Sources

  • 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $1$. Weierstrass-Bolzano Theorem
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