Math question?

Is the amount Is A = {(x,y,z, f(u)) in R^4 : u in Y and ||(x,y,z)|| = g(u)} a three-dimensional submanifold if z is a curve with z(u) = (f(u), g(u)) defined on an interval Y such that z(Y) is a one-dimensional submanifold? I'd say you can define a function h(x,y,z,t) = ||(x,y,z)|| – g(u)…

From how many non-repeating decimal places is a number considered irrational?

What is the guideline? For example, with pi, billions of decimal places are known, but nothing repeats itself. Therefore, it is assumed that it will continue like this forever, and so the number is irrational. From how many decimal places were determined did they first say that the number is irrational because nothing repeats itself.

Prove?

I want to prove this here, but it's obvious that it's true. Now I'm having a hard time proving it. Of course, addition and scalar multiplication hold, and from these follow linear combination and the so-called linear span, but how can I work with it now?