Let $\Gamma\subset \mathbb R^N$ be a $N-1$ rectifiable curve such that $\mathcal H^{N-1}(\Gamma)<\infty$. I am wondering that would it be possible to partition it into countably many connection pieces, up to $\epsilon>0$ error? (where $\epsilon>0$ is given)
That is, would it be possible to find a set $\{\Gamma_n\}_{n=1}^\infty$ such that each $\Gamma_n\subset \Gamma$ is connected, and $$ \mathcal H^{N-1}\left(\Gamma\setminus \bigcup_{n=1}\Gamma_n\right)<\epsilon. $$
My try: I simply think there can not be uncountably many connected pieces with positive $\mathcal H^{N-1}$ measure, otherwise $$ \sum_{n=1}^{\infty} \mathcal H^{N-1}(\Gamma_n) = \infty, $$ which contradicts to the fact that $\Gamma$ has finite measure.
Is it my argument correct?