Is a finite union of bounded sets bounded in any metrical space? In any metrical space $(M,d_M)$, consider $n$ bounded subsets $S_i\subset M$. Then, is $\cup_i^nS_i$ bounded? If so, why?

Since each $S_j$ is bounded, there exists a point $p_j$ such that $S_j\subset B(p_j,r_j)$. Now take $p=p_1$, $r=\max\\{r_1,\ldots,r_n\\}+\max_j\\{d(p_1,p_j)\\}$. If $x\in S_j$, then $$ d(x,p)\leq d(x,p_j)+d(p_j,p_1)\leq r_j+d(p_j,p_1)\leq r. $$ So $x\in B(p,r)$, and this shows that $S_j\subset B(p,r)$ for all $j$. Thus $$ \bigcup_j S_j\subset B(p,r). $$