# Prove that $H$ reduces to $H\varepsilon$

I have to prove that $$H_\varepsilon = \{ \mid M\ \text{halts on input }\varepsilon\}$$ reduces to $$H$$ (the halting problem).

I am very confused how to PROVE it, I mean it is clear that we can take a TM $$M$$ that decides $$H$$ and then build a TM $$M\varepsilon$$ that decides $$H\varepsilon$$ by taking the input $$$$ and simulating $$M$$ on input $$(, \varepsilon)$$, then accepting when $$M$$ accepts.

But how do I prove I can do that, how do I prove that this mapping from $$H$$ to $$H_\varepsilon$$ exists?

• Are you sure you want to reduce $H$ to $H\varepsilon$, not to reduce $H\varepsilon$ to $H$? – xskxzr Jun 16 '18 at 17:21