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: <math>\limsup_{n\to\infty}\Pr(X_n \in F) \leq \Pr(X\in F) \text{ for every closed set } F.</math>
Fix an arbitrary closed set ''F''⊂''S′''. Denote by ''g''<sup>−1</sup>(''F'') the pre-image of ''F'' under the mapping ''g'': the set of all points ''x'' ∈ ''S'' such that ''g''(''x'')∈''F''. Consider a sequence {''x<sub>k</sub>''} such that ''g''(''x<sub>k</sub>'') ∈ ''F'' and ''x<sub>k</sub>'' → ''x''. Then this sequence lies in ''g''<sup>−1</sup>(''F''), and its limit point ''x'' belongs to the [[closure (topology)|closure]] of this set, <math display="inline"> X\in\overline{g^{-1}(F)} </math> <!-- <span style="text-decoration:overline">''g''<sup>−1</sup>(''F'')</span> --> (by definition of the closure). The point ''x'' may be either:
* a continuity point of ''g'', in which case ''g''(''x<sub>k</sub>'') → ''g''(''x''), and hence ''g''(''x'') ∈ ''F'' because ''F'' is a closed set, and therefore in this case ''x'' belongs to the pre-image of ''F'', or
* a discontinuity point of ''g'', so that ''x'' ∈ ''D<sub>g</sub>''.
Thus the following relationship holds:
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