Turing machine with "move to origin" instead of "move left" You have a Turing machine which has its memory tape unbounded on the
  right side which means that there is a left most cell and the head cannot move left beyond it since the tape is finished. Unfortunately, you also find that on execution of a head move left instruction, rather than moving to the adjacent left cell, the head moves all the way back to the initial left most cell of the tape. Now figure out whether you can still use this TM effectively. The Turing machine with left initialize is similar to an ordinary Turing machine, but the transition function has the form
  
  $$\delta  \colon Q × Γ → Q × Γ × \{R, \mathit{INIT}\}.$$
  
  If $\delta(q, a) = (r, b, \mathit{INIT})$, when the machine is in state $q$ reading an $a$, the machines head
  jumps to the left-hand end of the tape after it writes $b$ on the tape and enters state $r$. Show
  that you can program this TM such that it simulates a standard TM. 


I can't figure out how to simulate this as standard TM. One thought I have is to copy the content of the tape which is afterwards the left move to the starting point of the tape before making a left move. Any further help would be appreciated.
When you want to move one position to the left, execute the following algorithm:


Mark the current position as special.
Move to the origin.
Translate the tape one cell to the right, while keeping the special marker in place.
Move to the origin.
Scan until reaching the special marker.
If you move to the right, don't forget to erase the special marker.


(This simulates a Turing machine with a single two-sided tape.)