In the proof that every type 1 grammar can be accepted by some TM, we simulated the productions of the grammar by a series of DELETEs followed by a series of INSERTs.
(i) Show that if the grammar being simulated were context-sensitive, the working string simulation field would never be larger than the input itself.
(ii) Show that this means that the total length of the section of the TM T_APE being used in the simulation reaches a maximum of 2n + 2 cells, where n is the length of the input string. This is a simple linear function of the size of the input. This is what is meant by the terminology "linear bounded automaton."