Suppose we have $$f_{m}(n) = \frac{1}{n}g_{m}(n) + o(\frac{1}{n})$$ where the little-o notation is uniform in the variable $m$ as $n \rightarrow \infty$. Under what conditions is $f(m,n) = o(\frac{1}{n})$? Under what further conditions (if any) is the little-o error for $f_{m}(n) = o(\frac{1}{n})$ uniform in $m$?

This problem is trying to ask us about what order $g_{m}(n)$ is. I think that $f_{m}(n) = o(\frac{1}{n})$ provided $\lim_{n \rightarrow \infty}g_{m}(n)$ exists. For the second part, I am not sure whether $g_{m}(n)$ needs to have uniformity in $m$ for its error.


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