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 $$\sup_M|D_0^m[g(T)-g(t)]|=\int\limits_t^TD_0^m[\tfrac{\partial}{\partial t}g]\to 0\ \ \text{as}\ \ t\to T-.$$ 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 $t\to T-$.