>>12497009See
>>12497010In some specific cases, a sequence can be described in finite terms. As an example, if a_n = n, and b_n = n + 1, then a_n * b_n = n * (n + 1) = n^2 + n.
This is perfectly fine since I have manipulated the symbols that represent the ability to generated an unending sequence of terms. I am not doing anything explicitly term-by-term since in this case, I have a general expression for the n-th term which allows me to circumvent the infinite amount of work.
However, this is not actually what a sequence is according to standard terminology. Sequences in general are modeled with choice functions, which means that if you want to manipulate them, you are forced to go term by term and do an infinite amount of work.
Also, more generally, if we consider S = [1, 2, 3, ...], which would be *all* the terms generated by the sequence a_n, you are trying to capture an unending process by just putting brackets or braces around the terms and claiming this is okay because of axiomatics.
