So I need some clarification of Ratcliffe's proof. Condensed information and question below.
STATEMENT: Before I proceed I will give some abbreviate notation. We may assume $c=\infty$. Furthermore, we have two sequences $f_ix_i$ and $f_ig_ix_i$ which converges to $a$ and $b$ ,respectively. If we let $R_i$ be the ray connecting $f_ix_i$ to $f_ig_ix$ then it is claimed that $R_i$ converges to the ray $(a,b)$, but its not clear rigorously why this is true.
QUESTION: Why does the sequence of rays $R_i$ converge to the ray $(a,b)$, where convergence is taken with respect to Hausdorff metric.