Using Closed/Open Balls in Rosenlicht's Intro to Analysis Proofs

[sampahmel]

Homework Statement

In Rosenlicht's Intro to Analysis, there is a proposition (p. 52).

A Cauchy sequence of points in a metric space is bounded.

Proof: For if the sequence is P1, P2, P3, ... and ε is any positive number and N an integer such tat d(Pn, Pm) < ε if n, m > N, then for any fixed m > N the entire sequence is contained in the closed ball of center Pm and radius of max{d(Pm, P1),..., d(Pm, PN), ε}

Homework Equations

So I wonder when do I use closed or open ball in the proof? I don't really see any difference. Can anybody shed some light on this matter.

 

[jbunniii]

So I wonder when do I use closed or open ball in the proof? I don't really see any difference. Can anybody shed some light on this matter.

For checking boundedness, it makes no difference which one you use. A sequence is bounded if it is contained in some ball with finite radius, and whether the ball is open or closed is irrelevant.

The author wrote "closed" in this case because with the radius he has chosen, the sequence is not necessarily contained in the open ball with that radius. However, if he had added any positive constant to his radius, then the sequence would have been contained in both the open and closed ball with the expanded radius.