The Hausdorff distance between sequences of sets

Question

Here is my problem. Consider 3 sequences of sets, for $n\rightarrow \infty$:

$$A_n=\{\frac{n-1}{n},\frac{n}{n+1}\}$$

$$B_n=(\frac{n-1}{n},\frac{n}{n+1})$$

$$C_n=[\frac{n-1}{n},\frac{n}{n+1}]$$

which sequence converges in the Hausdorff distance to $\{1\}$?

Given the definition, and in particular the part that defines convergence in terms of closures, to me it seems that only $B_n$ convergence to $\{1\}$.

But maybe I am confused with my intuition? Thanks for all hints!

Answer 1

Note that "every" point of $\{1\}$ is at distance $\le \frac1n$ from every point of $A_n$ and vice versa. Hence $d(A_n,\{1\})=\frac1n\to 0$.

The same works for $C_n$, so also $C_n\to\{1\}$.

This would also work for $B_n$ if we were to extend the notion of Hausdorff distance to non-compact sets in a reasonable way. However, you must be aware that this is a bad idea: the distance between $B_n$ and $C_n$ would then be zero, and we no longer have a metric.