Take an input x of length 2. If M is in L then on input x it cannot exceed the first place in the tape.
Now take an input y of length 2^(200)+13 (my favorite number, along with 17) whose first letter is the same as x first letter. If M is
in L then it cannot exceed the first place in the tape for that input, y, as well, because if it does, it will exceed the first place in the tape
for x (it has no way to know the difference between x and y based on first place only).
So, no matter what the input is, if M is in L its behavior is based on the first letter only and it cannot go beyond first place.
This can easily be checked by simulating M's behavior.