If $<a_n>_{n=1} ^∞$ be a sequence of positive real numbers such that $a_n → l$ as $n→ ∞$ then $(a_1a_2a_3...a_n)^{1/n}→ l$ as $n→∞$ I know the proof when for $l>0$ But want to know whether this result hold for $l=0$ as well or not . If it holds for $l=0$ also then kindly provide some hints or suggestions for proving this .
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See this https://math.stackexchange.com/questions/2608175/prove-that-lim-n-to-infty-a-1a-2-ldots-a-n-frac-1n-l-given-that-l/2608315#2608315 – VJunior Feb 06 '18 at 18:33
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Just use the arithmetic-geometric mean:$$\sqrt[n]{a_1a_2\ldots a_n}\leqslant\frac{a_1+a_2+\cdots+a_n}n.$$
José Carlos Santos
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