Let $C$ be the Cantor set with metric from $(\mathbb{R},|\cdot|)$ and $P = \prod_{k=1}^\infty \{0,\frac{1}{2^k}\}$ with metric from $(\ell_1, \lVert \cdot \rVert_1).$ Show that there exists a continuous bijection $f : C\to P.$
Every element of the Cantor set can be uniquely written in the form $\sum_{k=1}^\infty \frac{a_k}{3^k}, a_k \in \{0,2\}\forall k.$ Then it suffices to show that the function $f : C\to P, f(\sum_{k=1}^\infty \frac{a_k}{3^k}) = \prod_{k=1}^\infty \{\frac{a_k/2}{2^k}\}$ is continuous. I think one could show it's sequentially continous, but I'm not really sure about the details. I can't really show that it's continuous because each of its components is continuous, as it has infinitely many components. And I'm finding it really hard to choose the right $\delta > 0$. Is there another function for which it's easier to prove continuity?
