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==Proof <ref>This proof has been adopted from {{harv|van der Vaart|1998|loc=Theorem 2.3}}</ref>==
===Convergence in distribution===
We will need a particular statement from the [[portmanteau theorem]]: that convergence in distribution <math>X_n\xrightarrow{d}X</math> is equivalent to
: <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 arbitrary closed set ''F''⊂''S′''. Denote ''g''<sup>−1</sup>(''F'') its pre-image 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 point ''x'' belongs to the closure of this set. Point ''x'' may be either the continuity point of ''g'', in which case ''g(x<sub>k</sub>)''→''g(x)'', which must lie in ''F'' because ''F'' is a closed set, and therefore ''x'' belongs to the pre-image of ''F''; or ''x'' may be the discontinuity point of ''g'': ''x''∈''D<sub>g</sub>''. Thus we established the following relationship:
: <math>
\overline{g^{-1}(F)} \ \subset\ g^{-1}(F) \cup D_g\ .
</math>
Now let’s consider the event {''g(X<sub>n</sub>)''∈''F''}. Then
: <math>
\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 lim sup 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
\operatorname{Pr}\big(X\in g^{-1}(F)\cup D_g\big) \leq \\
& \operatorname{Pr}\big(X \in g^{-1}(F)\big) + \operatorname{Pr}(X\in D_g) =
\operatorname{Pr}\big(g(X) \in F\big) + 0.
\end{align}</math>
Plugging back into the original expression, we see that
: <math>
\limsup_{n\to\infty} \operatorname{Pr}\big(g(X_n)\in F\big) \leq \operatorname{Pr}\big(g(X) \in F\big),
</math>
which by the portmanteau theorem implies that ''g(X<sub>n</sub>)'' converges to ''g(X)'' in distribution.
===Convergence in probability===
Fix arbitrary ''ε''>0. Then for any ''δ''>0 consider the set ''B<sub>δ</sub>'' defined as
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