I came across the following example that proves that the blank tape halting problem is not decidable.I understand the proof technique, but I just don't see how the blank tape problem is shown to be undecidable in the proof.It says $M_{w}$ starts with a blank tape but then writes $w$ onto the tape, so there is no difference to the general halting problem.How does this imply that the blank tape halting problem is not decidable? I can't find the logical conclusion. Couldn't we just replace the blank tape halting problem with another DECIDABLE problem and argue the same way?
Understanding the proof of the halting problem
computability;proof techniques;undecidability;halting problem