How could I prove that L1 is undecidable?

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The Problem:

Let L1 = {Kod(M) | M is a TM and when started with a blank input tape, M
eventually writes a 1 somewhere on the tape } Prove that L1 is undecidable.


I have the following 2 TMs whcih I have proved are undecidable to help me prove the current problem is undecidable. I want to use the reduction method to do so.

Key:

  • TM : turing machine
  • w : input string for the TM
  • < kod(m) > : just means a binary encoding of a turing machine m being that TM.

    Ltm = { < kod(m), w > | m is a TM and m accepts w }
    
    Lh = { < kod(m), w > | m is a TM and m halts on w }