The assumption is that $\{r_1,\dots,r_k\}$ is bounded, say by $M$ above and $-M$ below.

Now consider the set $S = \{r_1,\dots,r_k,r_{k+1}\}$. By assumption we have $|r_j| \leq M$, for $j = 1,\dots,k$.

Let $M' = \max\{M,|r_{k+1}|\}$. Show that for any $r_j \in S$, we have $|r_j| \leq M'$.