Prove that there exists a machine which decides an infinite subset of halting problem
Let $M$ be a TM which accepts all inputs. Then $\{ \langle M, w \rangle \mid w \in \{ 0, 1 \}^\ast \}$ is an infinite subset of $H$ and (easily) decidable.

Going further, let $M_w$ be a TM which halts if and only if its input is not equal to $w \in \{ 0, 1 \}^\ast$ and let it be such that $\langle M_w \rangle$ is computable from any $w$ (e.g., $w$ is a substring of the description of $M_w$). Then the set $\{ \langle M_w, w' \rangle \mid w, w' \in \{ 0, 1 \}^\ast \}$ is another infinite subset of $H$ which is decidable. The description for a decider could be, for example: Given $\langle M_w, w' \rangle$, compute $\langle M_{w'} \rangle$ and accept if and only if it is equal to $\langle M_w \rangle$.

A warning: in the question text you referred to "a machine that decides any infinite subset of $H$". Indeed, no such machine exists (since $H$ is an infinite subset of itself); however, this is not what the question title (presumably the intended question) is implying.