How can 'infinity' be a viable mathamatical concept when one must consider the necessity of alternative dimensional 'inflection points' to even define infinity?

If a particular mathamatical conclusion is a simple line above a series of numbers that repeat indefinately, how can we know there is not a divergance after, for example, the 1,0000,0000,000,000th repettition of said numbers? And this variance or divergence would be the anamoly, or 'alternative dimension' expressed in numbers that is necessary to preserve the integrity of the definition (or concept) of infinity?

Wouldn't a variance in the very definition of infinity be required that would cross multiple dimensions?