Sounds like a home work problem?

Note that 
$$g(T)=\lim_{t\to T-}g(t)=g(0)+\int\limits_0^T\tfrac{\partial}{\partial t}g$$
Then you get 
$$|D_0^m g(T)|\le \mathrm{Const}(m)$$
and
$$\sup_M|D_0^m[g(T)-g(t_0)]|=\sup_M\int\limits_{t_0}^TD_0^m[\tfrac{\partial}{\partial t}g]\\,dt\to 0\ \ \text{as}\ \ t_0\to T-.$$
One can cover $M$ by charts with bounded $g(0)$-Christoffel symbols in each.
Then the above inequalities imply that $g(T)$ is $C^\infty$-smooth and 
$g(t)\to g(T)$ in $C^\infty$-topology as $t\to T-$.