But the "example from real analysis" section in your link uses exactly the same limiting technique to construct e. That section shows that 0.999... is constructible.

First we have to agree on what 0.999 is, then we can call it constructible or not.

Numbers formed by repeated 9's appended to 0.9 are certainly Turing computable. Which has the meaning that we have a terminating algorithm which, given a natural number N, will compute the N-th digit of the infinite sequence 0.999...

This is the same way that pi is computable. Given an N, we can compute the N-th digit of pi in a finite number of steps.

We don't say that 3.141... = pi! Unless, by convention, we agree that this "..." syntax has the semantics of (for instance) denoting the limiting numeric value of the non-terminating algorithm for producing the digits of pi ad infinitum. That is to say, "insert here a process for calculating the remaining digits of pi, and take this whole expression then to be the limiting value to which that converges".

But that's exactly what the ellipsis means. What other interpretation is there? I might as well argue that 1 = 1 only if we agree on the convention that = means "equals".