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: <math>\limsup_{n\to\infty}\operatorname{Pr}(X_n \in F) \leq \operatorname{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, <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:
: <math>
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\operatorname{Pr}\big(g(X_n)\in F\big) = \operatorname{Pr}\big(X_n\in g^{-1}(F)\big) \leq \operatorname{Pr}\big(X_n\in \overline{g^{-1}(F)}\big),
</math>
and by the portmanteau theorem the [[limsup]] of the last expression is less than or equal to Pr(''X'' ∈ <span style="text-decoration:overline">''g''<sup>−1</sup>(''F'')</span>). Using the formula we derived in the previous paragraph, this can be written as
: <math>\begin{align}
& \operatorname{Pr}\big(X\in \overline{g^{-1}(F)}\big) \leq
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On plugging this back into the original expression, it can be seen that
: <math>
\limsup_{n\to\infty} \
</math>
which, by the portmanteau theorem, implies that ''g''(''X<sub>n</sub>'') converges to ''g''(''X'') in distribution.
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