In Hilbert space $u$, Let $T_1$,$T_2$ is selfadjoint operator, if exit $c>0$ such that $cI\le T_1\le T_2$, prove $T_1$,$T_2$ have a bounded inverse operator and $c^{-1}I\ge T_1^{-1}\ge T_2^{-1}$.
I have proved $T_1$ and $T_2$ have inverse operators by using the relation between the range of the adjoint operator and the null space of the operator. How do I prove the second half of the problem and the boundedness of the inverse operator?