Assume that $\{C_i\}_{1\leq i\leq n}\subseteq \mathbb{R}$ is a collection of closed, bounded and non-empty sets in $\mathbb{R}$. Further assume that $\lim_{i\rightarrow \infty}diam(C_i) = 0$.
Prop: $\bigcap_{1\leq i\leq n} C_i$ contains exactly one point.
Pf: Assume that $\{x_i\}, \{y_i\}\subseteq \bigcap_{1\leq i\leq n} C_i$ are a Cauchy sequence such that $\{x_n\}\rightarrow x$ and $\{y_n\}\rightarrow y$. If $\lim_{i\rightarrow \infty}diam(C_i) = 0$, then $d(x, y) = d(\lim_{i\rightarrow \infty} x_i, \lim_{x\rightarrow \infty}y_i) = \lim_{i\rightarrow \infty} d(x_i, y_i) = 0$. Thus, $\bigcap_{1\leq i\leq n} C_i$ contains exactly one point.
Does my proof make sense?