Equivalence of Definitions of Tangent Vector

Definition
Let $M$ be a smooth manifold.

Let $m \in M$ be a point.

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

Define $C^\infty\left({V, \R} \right)$ to be the set of all smooth mappings $f : V \to \R$.

Then a tangent vector $X_m$ at $m$ is a linear mapping $X_m : C^\infty\left({V, \R} \right) \to \R$ satisfying the Leibniz law:
 * $\displaystyle X_m \left( {f g} \right) = X_m \left( {f}\right) \, g\left({m} \right) +f \left({m} \right) \, X_m \left( {g} \right)$

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

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
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 $f= \lambda$.