Let $\operatorname{dim}_{H}$ denote the Hausdorff dimension of a set in $\mathbb{R}^{n}$. Prove that if $\operatorname{dim}_{H}(F)<1$ then $F$ is totally disconnected.

[You may assume that if $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is a Lipschitz map then

$$\left.\operatorname{dim}_{H}(f(F)) \leqslant \operatorname{dim}_{H}(F) .\right]$$