Need hints or suggestions related to a following problem on sequences

Question

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 .

Answer 1

Just use the arithmetic-geometric mean:$$\sqrt[n]{a_1a_2\ldots a_n}\leqslant\frac{a_1+a_2+\cdots+a_n}n.$$