Let Zn, n 1, be a sequence of random variables and c a constant such that, for each 0, P Zn c 0 as n Show that, for any bounded continuous function g, E g(Zn) g(c) as n

Question: Let Zn, n 1, be a sequence of random variables and c a constant such that, for each > 0, P{|Zn c|

Let Zn, n ≥ 1, be a sequence of random variables and c a constant such that, for each ε > 0, P{|Zn − c| > ε}→0 as n→∞. Show that, for any bounded continuous function g,
E[g(Zn)]→g(c) as n→∞

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