$T-\lambda I$ is closed $\implies (T-\lambda I)^{-1}$ is closed

Question

Let X be a complex Banach Space ,$T:X\rightarrow X$ a linear operator, and $\lambda\in \rho(T)$(Resolvent set),If T is assumed to be closed ,then I have to prove that

$T-\lambda I$ is closed $\implies (T-\lambda I)^{-1}$ is closed

I'm proving a result of "Spectral theory of Normed Spcaes" there is a gap in my proof which requires

$T-\lambda I$ is closed $\implies (T-\lambda I)^{-1}$ is closed.

Please give any hint or suggestion to prove the above argument