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\Pr\big(g(X_n)\in F\big) = \Pr\big(X_n\in g^{-1}(F)\big) \leq \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 <math display="inline"> \Pr\left(X\in\overline{g^{-1}(F)} \right). </math> <!-- 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}
& \Pr\big(X\in \overline{g^{-1}(F)}\big) \leq
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