An important consideration that pops up in many fields of mathematics
, usually when one talks about the effect of an operation performed many times
. Frequently, statements hold true for an arbitrarily large but finite
number of operations, but not for an infinite number.
For example, any
finite sum of rational numbers is again a rational number. However, the infinite sum
4 - 4/3 + 4/5 - 4/7 + ...
, the probably best-known irrational number
On a more advanced level, the question of different orders of infinity, of countability arises. For example, a countable sum of positive numbers can converge (e.g. a geometric series), but an uncountable sum always diverges (Actually, it would be more correct to say that the traditional concept of convergence is tied to countability, though one can define it more generally, as done in topology).
It is important to understand this, not just for clean proofs, but also for clean problem statements: when writing down an infinite sum (or product, or union, or intersection), indexing it with something like "i = 1 ... n" implicitly assumes a finite operation, and "i = 1 ... infinity" implicitly assumes a countable operation. To allow for an uncountable operation, one has to use a general index set. Depending on the rest of the problem, assumed finiteness or countability could make a big difference.