Question related to asymptotic equivalence and possibly regularly varying functions

Question

Let $f(n)$, $n \in \mathbb{N}$, be a function such that $f(n) \sim n^{-1/2}$. Let $\delta \in (0,1)$. I want to prove that $$ \sum_{j \leq \lfloor \delta n \rfloor} f(j) \sim 2 \delta n \, f(\lfloor \delta n \rfloor). $$ I am fairly lost here. I thought about using that $f(n)$ is a regularly varying function, but got nowhere. Any ideas why this is true? (The parentheses $\lfloor \ \rfloor$ are supposed to mean taking the next lower integer, e.g. $\lfloor 5/2 \rfloor = 2$.)