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Oo (b) there exists a sequence (Xh) converging to x in X such that (sc- F)(x) ~ lim sup F(Xh) . h ..... oo Proof. 3 and from the inequality sc- F:S" F. To prove (b), we may assume (sc- F)(x) < +00. Let (Uk) be 30 An Introduction to r -convergence a countable base for the neighbourhood system of x such that U h+l ~ Uh for every hEN, and let (th) be a sequence converging to (sc- F)(x) in R such that th > (sc- F)(x) for every hEN. 3 for every hEN we have th > inf F(y) , hence there exists Xh E Uh such that th > F(Xh)' yEUh Then (Xh) converges to x in X and o which proves (b).

Let X = R, let (qh) be an enumeration of the set of all rational numbers and let 0, if x = qk for some k ~ h, Fh(X) = { 1, otherwise. 4. 7. If (Fh) is a decreasing sequence converying to F pointwise, then (Fh) r -converyes to sc- F. Proof. 2. S. We say that the sequence (Fh) is equi-lower semicontinuous at a point x E X if for every e > 0 there exists U E N(x) such that Fh(y) ~ Fh(X) - e for every y E U and for every hEN. We say that (Fh) is equi-lower semicontinuous on X if (Fh ) is equi-Iower semicontinuous at each point x EX.

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