Bolzano-Weierstrass Theorem/Lemma 2
Jump to navigation
Jump to search
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