prove that if f(x) is monotone increasing on an open interval (a,b) and x_0 is any point in (a,b), then lim_{x \to x_0+} f(x) exists and satisfies lim_{x to x_0+} f(x) >= f(x_0). (from the right side)

Is this proof correct? if not could someone please correct it/ explain what I'm doing wrong?

Fix in (a,b). First consider x in ( ,b] then f(x)>= f( ) so inf f(x) ( <x<b) >=f( )

Claim: lim f(x)=inf f(x) ( <=x<b) . Let S=inf f(x). Let >0. Then there exists x in ( ,b) such that f(x )<S- .

But if x is in (x , ) then f(x )>=f(x) so S<=f(x)<S-

Hence |f(x)-s|< for all x in (x , ).

therefore, lim f(x) (x-> +) exists and equals S since S>=f( ) we get lim (x-> +) >= f( )