What are toy examples of single-copy entanglement conversion?

In (Vidal 1999) they prove that, given any pair of pure bipartite states written as$$|\Psi\rangle = \sum_{i=1}^n \sqrt{\alpha_i}|i,i\rangle,\qquad |\Phi\rangle = \sum_{i=1}^n \sqrt{\beta_i}|i,i\rangle,$$with $\alpha_1\ge \alpha_2\ge\cdots\ge \alpha_n$, $\beta_1\ge \beta_2\ge\cdots\ge \beta_n$, the maximum probability of converting $|\Psi\rangle$ into $|\Phi\rangle$ by means of any local strategy is given by$$P(\Psi\to\Phi) = \min_{\ell\in[1,n]} \frac{\sum_{i=\ell}^{n}\alpha_i}{\sum_{i=\ell}^{n}\beta_i}.$$What are some toy examples to better understand the idea behind the strategy, and the optimality proof?