Arbitrary Intersection of Bounded Sets Is Bounded Let $\left\{C_i\right\}_{i\in I}$ be an arbitrary collection of bounded sets in $\mathbb R^n$ and consider $C=\bigcap_{i\in I}C_i$. Moreover, let each $C_i$ be bounded by $B_{r_i}$, where $B_{r_i}$ is the ball of radius $r_i$ centered at the origin.
It is often showed that $C$ is bounded by letting $r=\max\left\{r_i\right\}_{i\in I}$ and arguing that $C$ is bounded by $B_r$.
I do not buy this.
Consider the collection of bounded sets $\left\{B_{r_i}\right\}_{r_i>0}$. Then $r$ does not exist.
Am I missing something? If not, then may I be hinted at a different way of showing that $C$ is bounded?
This is my best argument so far: "Let $x\in C$. Then $x\in C_i$ for each $i\in I$. Because each $C_i$ is bounded, it follows that $x$ is bounded." However, I have a hunch feeling that it may be too "hand-wavy."

Does $B_{r_i}$ bound $C$ for some $i$?
