Equivalence of Definitions of Tangent Vector

Proposition
Definition 1 and Definition 2 in Definition:Tangent Space are equivalent.

Lemma 1
Let $X_m$ be a tangent vector at $m \in M$ according to Definition 1.

Let $V$ be an open neighborhood of $M$.

Let $f \in C^\infty\left( {V,\R } \right)$ be constant.

Then $X_m \left( { f}\right) = 0$.

Proof of Lemma 1
Let $f\left( { m}\right) = 0$.

Then, by constancy, $f= 0$ on $V$.

Hence, by linearity, $X_m(0)= 0$.

Let $f\left( { m}\right) \ne 0$.

$f$ is constant, iff $\exists \lambda \in \R : f( V )= \left\{ {\lambda} \right\}$, iff $f= \lambda$.