If it is undecidable, then that means no proof exists (that H will or will not halt.)
If no proof exists, then H will loop forever searching for one.
Therefore undecidability implies H will run forever. You've just proved this.
Therefore a proof exists that H will run forever (that one), and H will eventually find it.
Paradox...
As people have been saying, if H can make this argument it is inconsistent and does not work properly (i.e. it does not return True or False in the correct situations.)
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should be posted in Discussion, and not Main.
4. Open Threads should start on Monday, and end on Sunday.