Let $(X, τ)$ be a topological space. We have the following:
a.) If $F$ is closed, then $\operatorname{int}(\operatorname{cl}(\operatorname{int}(F)))=\operatorname{int}(F)$.
b.) From (a) you can deduce that for any open set $U$, $\operatorname{cl}(\operatorname{int}(\operatorname{cl}(U)))=\operatorname{cl}(U)$
if you set $U:=\operatorname{int}(X)$ you have the desired result.
$\operatorname{Int}(\operatorname{Cl}\operatorname{Int}X) ⊂ \operatorname{Cl}\operatorname{Int}X \Rightarrow \operatorname{Cl}(\operatorname{Int}\operatorname{Cl}\operatorname{Int}X) ⊂ \operatorname{Cl}(\operatorname{Cl}\operatorname{Int}X) = \operatorname{Cl}\operatorname{Int}X,$
$\operatorname{Cl}(\operatorname{Int}X) ⊃ \operatorname{Int}X \Rightarrow \operatorname{Cl}\operatorname{Int}(\operatorname{Cl}\operatorname{Int}X) ⊃ \operatorname{Cl}\operatorname{Int}(\operatorname{Int}X) = \operatorname{Cl}\operatorname{Int}X$