Infinite product of limit elements.

Question

Suppose I have an array of strictly positive elements $\{a_{k}^{m}\}_{k,m\in\mathbb{N}}$ such that

1. $$ a_k:= \lim_{m\rightarrow\infty}a_{k}^{m} > 0 $$
2. $$\lim_{m\rightarrow\infty}\prod_{k=1}^{m}a_{k}^{m} = 0 $$

Is convergence of $a_{k}^{m} \rightarrow a_k$ uniform over $k$ enough to ensure that $$ \lim_{m\rightarrow\infty}\prod_{k=1}^{m}a_{k} = 0 . $$

Answer 1

Uniform convergence of $a_k^m \to a_k$ is not sufficient. Consider

$$a_k^m = 1 - \frac{1}{\sqrt[3]{k+1}\sqrt{m+1}}.$$

Then $a_k^m \to 1$ uniformly, but

$$\sum_{k = 1}^m \log a_k^m < -\sum_{k = 1}^m \frac{1}{\sqrt[3]{k+1}\sqrt{m+1}} \sim -\frac{3}{2} \cdot \frac{(m+1)^{2/3}}{\sqrt{m+1}} = -\frac{3}{2}(m+1)^{1/6} \to -\infty,$$

so

$$\lim_{m\to\infty} \prod_{k = 1}^m a_k^m = 0.$$