# Combining little-o notation

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.