Is it always possible to have a (local) time coordinate in GR?

Let $(M^{n+1},g)$ be a Lorentzian manifold. Given $p\in M$, we will show that there is a coordinate system $(x^\mu)$ defined on an open set $p\in U\subset M$ such that $\partial_0$ is a timelike vector field, and $\partial_i$ are spacelike vector fields for $i=1,\dotsc,n$.

Let $(x^\mu)$ be an arbitrary chart defined on $U\ni p$. It is known that $T_pM$ is the span of $\{\partial_0,\partial_1,\dotsc,\partial_n\}$. As $g_p$ has signature $(-,+,\dotsc,+)$, we may find linearly independent vectors $v_\mu$, $\mu=0,1,\dotsc,n$, such that $g_p(v_0,v_0)=-1,$ $g_p(v_i,v_i)=+1$. These vectors are linear combinations of $\{\partial_0,\partial_1,\dotsc,\partial_n\}$. By a linear change of coordinates, we can find a coordinate system $(y^\mu)$ such that $\partial/\partial y^\mu=\partial_\mu'=v_\mu$ at $p$. By continuity, there is a neighborhood $V_0\subset U$ of $p$ such that $g(\partial_0',\partial_0')<0$, i.e., $\partial_0'$ is timelike on $V_0$. Similarly, there exist neighborhoods $V_i$ such that $g(\partial_i',\partial_i')>0$ on $V_i$. We take $V=V_0\cap\cdots \cap V_n$, which is a neighborhood of $p$. By changing each coordinate value by a constant, we can adjust the origin without changing the aforementioned vector fields. Then $(y^\mu)$ is the desired coordinate system on $V$.