Is .9 repeating an approximation of 1?

Or is .9 repeating, in fact, = 1?

The proofs (read the Wiki, as much as I despise Wiki) all tell me that it is...but they never satisfy the question as to how something that continues to the infinite but remains shy of the finite is the same as the definition of the finite.

The argument was that 1.9 repeating (though there is no reason .9 repeating wouldn't work) is 2 (or in your argument, 1).

.9 repeating = 1

I can see the approximation resulting from the infinitesimal. I cannot see the fact.