Unrecognizability of HALT I am currently stuck on a question from Sipser's book of automata. The question goes like this:


  Consider the following language: V = {w|w = 0a for some a which is an
  element of HALT} U {w|w=1b for some b which is an element of
  complement of HALT }
  
  Prove that V is unrecognizable? Prove that complement of V is
  unrecognizable?


I kinda know that to prove them we have to use contradiction but I got stuck after that and haven't been able to come up with any solution up till now. Any pointers/hints would be appreciated.
If $V$ was recognizable then the language of Halting problem could be reduced to $V$ as following.

Let $M_V$ be a TM recognizing $V$, i.e, $M_V$ accepts $x$ if $x \in V$. Let $u$ be some string (description of a TM). Then we run $M_V$ on $0u$ and $1u$ by dovetailing. $M_V$ will eventually halt on one of these inputs since $u$ is either in $HALT$ or not in $HALT$. So, when it halts we simply check: if $M$ has halted on $0u$ then $u \in HALT$, else if $M$ has halted on 1u then $u \notin HALT$. Thus this implies that $HALT$ is decidable which is impossible. 

Similarly for $\overline{V}$. Let $u$ be some string (description of a TM). If $M_{\overline{V}}$ accepts $0u$ then $u \notin HALT$. If $M_{\overline{V}}$ accepts $1u$ then $u \in HALT$.