@Dr.Peterson, good call!!
You show me: try to prove your claim! First you need to say exactly what this means.
Let n be an irrational number such that n =t.abcd... (a,b, c, etc. are its digits)
n's rational approximation with a precision to the 10ths decimal place is ta/10
n's rational approximation with a precision to the 100ths decimal place is tab/100
so on and so forth
Archimedes use
the method of exhaustion to approximate
π. I believe it's a
general solution we could use for any irrational number.
We also have approximations using sums of series (re Leibniz's formula for
π and also there's a convergent series sum for
e (Euler's number). The summation can be terminated at any arbitrary point, determined by the level of precision one desires, oui?
You can state it; but can you prove it?
You're implicitly claiming that there is a mapping from each irrational to one approximation. I say there is not.
Non liquet, but

, you need to disprove me now.
If you claim that there is a unique rational approximation for any irrational number -- that means, there is only one -- then you have to show how it is determined. You have done no such thing.
The (x, y) pair, where x is the irrational and y the rational approximation, is unique.
There are an infinite number of rational approximations for any irrational number (the desired precision decides the satisfactoriness of the approximation). For example
722 as an approximation for
π is accurate only to the 2nd decimal place, while
115315 (Zu's constant) is more accurate.
Any given approximation will be ambiguous so long as the digits of the irrationals it approximates are identical, but they can't all be identical, otherwise they would be the exact same number. So, to disambiguate we use the digit that's different to construct different rational apprximations. So a unique pairing between irrationals and rationals (their approximation) is possible.
P.S. Why don't we use
100314=50157 as an approximation for
π?