TM with double-infinity tape semi-decides the same languages as classical TM
The intent of this assignment seems to be to demonstrate that allowing Turing Machines to treat the tape as infinite in both directions will not increase the computational power of the Turing Machine; in other words, that for any unbounded-tape TM, there exists an equivalent classical TM. This means you can't just &quot;reject if we go too far to the left&quot; on the unbounded-tape TM, because our proof has to be applicable to all unbounded-tape TMs, not just ones that conveniently do this.
Understanding the equivalence of these two definitions of TM is important because both definitions are widely used in the academia, and many proofs using TMs are easier on one of these models over the other. Knowing that they are equivalent allows one to pick the easier model for a given situation.
For a basic idea of how to prove these two models of computation equivalent, consider that shifting the entire contents of the tape to the right can be done by a classical TM, allowing it to dynamically claim more space on the left of existing tape contents as needed. You can utilize this behavior to simulate an unbounded-tape TM in a classical TM.