I am trying to prove a limit theorem. In one of the steps, it is written that, setting $$\Delta_n=\frac{V_1+...+V_r}{\sigma\sqrt{n}}-\frac{W_1+...+W_r}{\sigma\sqrt{n}},$$ where $V_1, ..., V_r$ are not independent random variables, but $W_1, ..., W_r$ are (I don't know if this fact is even used), then $$P(|\Delta_n| > \varepsilon) \leq \sum_{j=1}^r P\left(|V_j-W_j|>\frac{\varepsilon \sigma \sqrt{n}}{r} \right).$$
I am not being able to prove this inequality, and I feel like I am missing something really obvious. Thank you for your help.