Double sum that grows at sublinear rate

Question

Is there an example of a non-zero function $f: \mathbb{N} \to \mathbb{R}^+$ such that for any $n \in \mathbb{N}$, the following term is sublinear (or $o(n)$)?

$$\sum_{j=1}^n \sum_{i=1}^j f(i)$$

Answer 1

Work the other way around: You want:

$\begin{align*} \sum_{1 \le j \le n} s(j) = o(n) \end{align*}$

Here $s(j)$ is your inner sum. This means $s(j) = o(1)$. This requires in turn:

$\begin{align*} \sum_{1 \le i \le j} f(i) = o(1) \end{align*}$

But this is impossible, as the last sum is at least it's first term, it can't go down as $j$ increases.