Is 2 a number?

Is 4 a number?

Is 4/2 a number?

Is 3 a number?

Is 3/2 a number?

etc...

All of these symbols represent precise points on the numberline. Pi also represents a precise point on the numberline, so is it not a number?

Ironically, that response runs into the standard problem that many "limit" arguments have.

Generally speaking just because something looks like it's converging from some angle, it doesn't mean that it actually has a well-defined limit, and if it does then it also does not mean that the limit shares the properties of the items in the sequence of which it is the limit.

Examples: 1/n is strictly positive for all n. Its limit for n going to infinity, while well-defined, is not strictly positive. Another example: You can define pi as the limit of a sequence of rational numbers. But it's not rational itself.

So, no, your argument does not prove that pi is a number.

(I'm not arguing that pi is not a number. It definitely is. It's just that the argument is a different one.)