You can put a strictly equal sign and use three dots. Furthermore, no one claims that sqrt 7 is not what ppauper's link (or my solution for that matter) shows. The rest of your post suggests to me that you don't understand continuous fractions...

I see [$]\sqrt7[$] as a special case of my extremely popular and informative [$]e^5[$] thread (most viewed in Brainteaser). In that thread we used a bunch of methods, e.g. fixed-point contraction.

[$]y = \sqrt 7[$]

[$]y^2 = 7[$]

Solve by the sequence

[$] a_{n+1} = 1/2 (a_{n}+ 7/a_{n}) \enspace n >= 0[$] The term [$]a_{0}[$] is arbitrary.

Now

1. Prove { [$]a_{n}[$] } is a Cauchy sequence (for T4A).

2. Is the space complete? Is [$]\sqrt 7[$] a rational number?

3. Compute [$]y = \sqrt 7[$] by hand to two decimal positions accuracy. And make it snappy.

*you don't understand continuous fractions...*
It's a lost art.