Upper bound for the sum of positive numbers

Question

Let $\{X_i\}_{i=1}^{N}$ be a set of positive numbers and $\epsilon > 0$ such that $$ \sum_{i=1}^{N}{\frac{1}{X_i}} \geq \epsilon$$ Is it possible to find an upper bound such that $$ \sum_{i=1}^{N}{X_i} \leq f(\epsilon)$$ where $f$ is a smooth function.

Answer 1

No. For a given $\epsilon > 0$ you can choose $X_1 = 1/\epsilon$ and arbitrary positive $X_2, X_3, ...$. Then $\sum_{i=1}^{N}{\frac{1}{X_i}} \geq \epsilon$ is satisfied and $\sum_{i=1}^{N}{X_i}$ can be arbitrarily large.

Note that the inequality between the harmonic and the arithmetic mean provides an estimate in the opposite direction: $$ \sum_{i=1}^{N}{\frac{1}{X_i}} < \varepsilon \Longrightarrow \sum_{i=1}^{N}{X_i} > \frac{N^2}{\varepsilon} $$