Prove that var(min(X,c))<= var(X) where c is any constant and X is any random variable.?

Sol
Var(X)>=0
(1) X>=c
min(X，c)=c
Var[min(X，c)]=Var(c)=0
Var[min(X，c)]<=VAR(X)
(2) X<c
min(X，c)=x
Var[min(X，c)]=Var(X)
So
Var[min(X，c)]<=VAR(X)