[new2628]

This is a fine and defensible (although not mainstream) viewpoint with several adherents among mathematicians and various levels of success in making it formal and precise.

See e.g. https://en.wikipedia.org/wiki/Finitism

Doron Zeilberger would like to pat you on the back.

[baddox]

How do finitists describe the decimal representation of the fraction 1/3?

[improv32]

I don't think finitists have a problem with convergent sequences, they just wouldn't talk about infinities, rather than extending the sequence gets you as arbitrarily to 1/3 as you wish to go