Consider a sequence $(\varepsilon_n)_{n \in \mathbb{N}}$ of real numbers strictly greater than $-1$, convergent with limit zero. Deduce that:
$$\lim_{n \rightarrow +\infty} \sum_{i=1}^{n-1} \frac{1}{\sqrt{i(n-i)}} \cdot \left(\frac{(1+\varepsilon_i)(1+\varepsilon_{n-i})}{1+\varepsilon_n} - 1\right) = 0.$$