Consider an integer n which is the output of a URM program R with input 0, i.e. R(0) n. We might expect some positive integers to be the output of very short URM programs while others may require longer programs to output them. Let us make a (non-standard) definition: define the length of a positive integer n to be the the number of instructions in the shortest URM program R such that R(0) n. (There may be several different shortest programs of the same length, but thats OK. Also note the reason that we insist the input is zero is because we want to think of the length of n as a measure of how difficult it is to make the number n starting from nothing, i.e. zero).

Explain why we must always have length(n) n.