"Parameterizations are non-unique", how can we prove it?

Question

$$ \textbf{"Parameterizations are non-unique"} $$ I have seen this statement in several books and at Wikipedia.

However, I have never seen a proof of the statement. How can we prove it?

Answer 1

Actually, what it says is that they are generally nonunique. But if you have a parametrization of a curve $\gamma\colon[a,b]\longrightarrow\mathbb{R}^n$, you can always define, say,$$\begin{array}{rccc}\gamma^\star\colon&[a+1,b+1]&\longrightarrow&\mathbb{R}^n\\&t&\mapsto&\gamma(t-1)\end{array}$$and that's another parametrization.