Proving that the language of TMs with finite left head moves is undecidable
Hint: Reduce from the halting problem. Given a Turing machine $M$ and an input $w$, you want to come up with a new pair $\langle M',w \rangle$ such that $M$ halts on $w$ iff $M'$ makes a finite number of left moves on $w'$. Now if $M$ halts on $w$ then in particular it makes a finite number of left moves, but the other direction is not true. Think of a way of ensuring that if $M$ makes an infinite number of moves in any direction, then $M'$ makes an infinite number of left moves. (You might want to add some spurious moves.)