# Bolzano-Weierstrass property

• February 13th 2011, 07:58 AM
jacobn
Bolzano-Weierstrass property
Why would the interval [0, infinity) not have the Bolzano-Weierstrass property, which states that A set of real numbers E is closed and bounded if and only if every sequence of points chosen from the set has a subsequence that converges to a point that belongs to E?
• February 13th 2011, 08:11 AM
Plato
Quote:

Originally Posted by jacobn
Why would the interval [0, infinity) not have the Bolzano-Weierstrass property, which states that A set of real numbers E is closed and bounded if and only if every sequence of points chosen from the set has a subsequence that converges to a point that belongs to E?

Is the set $[0,\infty)$ bounded?

Let $\{x_n=n:n\in\mathbb{Z}^+\}$.
Does that sequence have a convergent subsequence?