I am having difficulty in reducing non halting problem to the given problem to prove that language is non R.E
For eg Completeness problem of TM .
In the above problem, we accept $\Sigma^*$ if H does not halt on w . For doing that, we give an input $x$ and simulate H on w for $|x|$ steps. I don't understand how can we deduce that H does not halt on w by just simulating H on w for $|x|$ steps.
Suppose H halts on w after $k+2$ steps and $|x| = k$. Now if we simulate H on w only for $k$ steps, how can we say that H will never halt or does not halt on w? It does not halt after $k$ steps but halts after $k+2$ steps.
I think that non halting problem is whether H does not halt on w and not whether H does not halt on w in k steps.
Can someone please help me in understanding the reduction?