Understanding Array Elements: n^2 vs. 8*n

[ehrenfest]

Homework Statement

For the problem at this site http://www.kalva.demon.co.uk/putnam/psoln/psol859.html, how an array with n^2 elements contain 8*n elements (8 for each positive integer) when n is not equal to 8? Does that type of algebra not work with an infinite number of elements...?

Homework Equations

The Attempt at a Solution

 

[Dick]

The point is a_{ij}>i*j. You will run out of numbers at a finite point even if a_{ij} gets to be hugely large. It's not really a Cantor problem. The index set is the limit.

 

[ehrenfest]

The point is a_{ij}>i*j. You will run out of numbers at a finite point even if a_{ij} gets to be hugely large. It's not really a Cantor problem. The index set is the limit.

My point is that if you think of the number of elements in the array as

$$lim_{n\to \infty} n^2$$ then it at least seems odd that this number could be the same as $$lim_{n\to\infty}8n$$.

So, you are saying that logic only holds for finite sets, right?
That is probably just my ignorance of infinite set theory.

 

[Dick]

The point is that you can find an N such that the number of pairs (i,j) with i*j<N is greater than 8*N. You could compute this N, if I'm doing my numbers right it's less than 10000. That means the problem doesn't have much to do with infinite set theory.