Gaussian curvature strictly positive and curvature of a curve contained in a surface

Question

this is my question.

"If S is a surface which K$>0$ (Gaussian curvature) then any curve contained in that surface has positive curvature (this curvature is the one for curves in $R^{3}$ )" (when I say positive I mean $>0$)

I think is true, maybe can I use the fact that if a surface contains a straight line then K$<=0$ for the points of that curve?