What are those numbers near either sides of 1?
0.999... and 1.000..1 do not exist? Because every number approaching 1 is 1?
So, all numbers very close to π is π?
(sin(x+h)-sin(x))/h is very close to cos(x), thus, is EQUAL to cos(x).
e^k= lim(x->∞) (1+k/n)^n ... Woh! rule of exponential arithmetic re-defined!

A[0]=0
A[n]=(A[n-1]+1)/2
What is A[n] when n approaches infinity? Oh! 'when' is invalid! 'infinity' is
a concept, not a number!
But, A[∞] will? get to 1, ℝ is dense, density property is not broken.

A transcendental number is a number that is not algebraic, not the kind of
number +-*/ can yield. But ℝ is closed and complete, Oh! where are all those
irrational numbers gone?

PS. I believe TM (program,algorithm) should/would be the foundation of math..
These are for programmers.