Editing Constructive analysis (section)

====Variations====
The above definition of <math>x\cong y</math> uses a common bound <math>\tfrac{2}{n}</math>. Other formalizations directly take as definition that for any fixed bound <math>\tfrac{2}{N}</math>, the numbers <math>x</math> and <math>y</math> must eventually be forever at least as close.
Exponentially falling bounds <math>2^{-n}</math> are also used, also say in a real number condition <math>\forall n. |x_n-x_{n+1}|<2^{-n}</math>, and likewise for the equality of two such reals. And also the sequences of rationals may be required to carry a modulus of convergence. Positivity properties may defined as being eventually forever apart by some rational.

[[axiom of non-choice|Function choice]] in <math>{\mathbb N}^{\mathbb N}</math> or stronger principles aid such frameworks.