It is known that When X is pseuodocompact then for any metric on Y it will do but now in question instead of X being pseuodocompact metric on Y is bounded.

Actually if one can answer this question in affirmative then very beautiful result can be proved

Fine topology on C(X,Y) where X is a topological space and Y is a metric space with metric 'd' is generated by a base consisting of sets of the form { B(f,t): f belongs to C(X,Y) and t is a continuous function on X whose range is a subset of positive real numbers}. And B(f,t) ={g belongs to C(X,Y): d(f(x),g(x))< t(x) for all x in X}