Show that the tangent bundle $TM $ of a variety $M$ is Hausdorff. I read a Lemma Introduction to smooth manifolds" which says that given a smooth n -manifold , then the tangent bundle  is a smooth 2-manifold, how can I prove this?

Let $v$ and $w$ be two distinct points in $T(M)$.  Say that $v \in T_x(M)$ and $w \in T_y(M)$ for some $x, y \in M$.  If $x = y$, then $v$ and $w$ can definitely be separated by an open set in $T(M)$, since locally $v$ and $w$ are given by the last $n$-coordinates in $\mathbb R^n \times \mathbb R^n$.
If $x \neq y$, choose disjoint charts $U$ and $V$ containing $x$ and $y$, respectively.  We can do this because $M$ is Hausdorff.  Then
$$\bigcup\limits_{p \in U} T_p(M) \cong U \times \mathbb R^n$$
$$\bigcup\limits_{p \in V} T_p(M) \cong V \times \mathbb R^n$$
are disjoint open sets of $T(M)$ containing $v$ and $w$, respectively.
