Consider the power set of , that is, the set of all possible integer sequences , where . If only infinite sequences are allowed, then these are homeomorphic to the interval and therefore uncountable.
Proof: Define . A visual way of looking at it is by expressing "in" and "out" of the set as bits in a binary string. Take the primes for example, , so . Therefore, any subset of can be expressed uniquely by a real number , converges regardless of a, , and .
*Note: A terminating decimal .100101 can be expressed uniquely as a nonterminating decimal .100100111111111... by properties of decimal expansions. However, it is counterintuitive to think that the set is the same as . But if we restrict ourselves to infinite subsets, we circumvent this problem. Besides, if the set of infinite subsets of is uncountable, than so certainly is the power set of .
I am not sure how rigorous I've been, but is this a correct statement? The set of all possible ordered integer sequences bounded below by 1 is homeomorphic to the interval (0,1] and therefore the power set of is uncountable.
Is this way of interpreting infinite sequences known and/or applicable in math?