Prove that $H$ reduces to $H\varepsilon$
This can be done by hard-coding a pre-processing stage to paste down $x$ to the input tape, before actually running $M$ on $x$ as input.

So given an instance $&lt;M,x&gt;$ of HALTING, you transform $M$ to $M_x$ that submerges the input $x$ into its initial stage. This can be done efficiently.

For example, if $x=010$, then start from the starting state $q$ of $M_x$ at left-marker $\$$, it moves one step to the right, and changes to state $q_0$. Then, $M_x$ prints out $0$, moves the head one step to the right and changes to state $q_1$. Then, $M_x$ prints out $1$, moves the head one step to the right and changes to state $q_2$. Then, $M_x$ prints out $0$, moves the head one step to the right and changes to state $q_3$. Then, $M_x$ moves all the way left back to the left-marker $\$$. Then, $M_x$ changes to the very starting state $q_M$ of $M$ and passes the control to $M$. Now, $M$ receives $x$ as input on the tape.

The above behavior of $M_x$ should be coded as its transition table. The description of $M_x$ is the produced instance of $H_\varepsilon$.

Finally, $M$ halts on $x$ iff. $M_x$ halts on $\varepsilon$.