Is For Only Finitely Many Indices j Justifiable in Compactness Proofs?

[ehrenfest]

Homework Statement

http://books.google.com/books?id=th...ts=yfNwPI7lv_&sig=6MWgB7WBS3P8NZ_a_7Dq_um7EvI
Can someone explain there rational for the statement "for only finitely many indices j" in the second paragraph of the proof?

Homework Equations

The Attempt at a Solution

 

[Hurkyl]

It means the set of j's such that ______ holds is finite.

 

[ehrenfest]

Sorry, I asked the wrong question. I wanted to ask how do they make the conclusion that there "is a grand total of only finitely many positive integers"?

 

[Hurkyl]

How many y's are there?
How many balls are there?
How many y's per ball?

 

[ehrenfest]

Of course! Finite times finite = finite.