This question is from my assignment in manifolds and I am following the book introduction to smooth manifolds.
Question: Let X,Y and Z be vector fields on $\mathbb{R}^n$. Let $\sigma : I \to \mathbb{R}^n$ be the integral curve of X starting at 0, where I is some open interval containing 0. If $Y(\sigma(t))= Z(\sigma(t))$ for all $t\in I$, show that $[X,Y](0) = [X,Z](0)$.
Simplifying using the definition of Lie brackets I get: $[X,Y](0) = [X,Z](0)= (X) (Y(0)- Z(0)) = (Y-Z) (X(0))$.
But the problem I am facing is that I don't know how use $Y(\sigma(t))= Z(\sigma(t))$ and thus I am not able to move foreward.
Can you please give me hints on how to use the information given in the question?
I shall be really thankful.