His argument works by exclusion; he excludes every notation that doesn't work with just the natural numbers.
Using strict inequality for the lower bound means you can't express sequences starting at 0 without expressing your lower bound as a negative number, which is ugly. So he prefers non-strict inequality for the lower bound.
Using non-strict inequality for the upper bound causes a similar problem. Consider the sequences (non-strict inequality on both sides) [0..2], [0..1], [0..0]. The first has three elements, the second two elements, the third one element. If you want to express a sequence with no elements starting at 0, that is impossible without using a negative number for the upper bound ([0..-1]). So he excludes using non-strict inequality for upper bounds.
What we're left with is non-strict inequality for the lower bound and strict inequality for the upper bound. This way we never have to leave the set of natural numbers to express natural number sequences, and as an added bonus, the difference between our bounds is equal to the length of the sequence.
Note that this part of the argument doesn't touch on 0 vs 1 yet, that is argued later in the essay.
Using strict inequality for the lower bound means you can't express sequences starting at 0 without expressing your lower bound as a negative number, which is ugly. So he prefers non-strict inequality for the lower bound.
Using non-strict inequality for the upper bound causes a similar problem. Consider the sequences (non-strict inequality on both sides) [0..2], [0..1], [0..0]. The first has three elements, the second two elements, the third one element. If you want to express a sequence with no elements starting at 0, that is impossible without using a negative number for the upper bound ([0..-1]). So he excludes using non-strict inequality for upper bounds.
What we're left with is non-strict inequality for the lower bound and strict inequality for the upper bound. This way we never have to leave the set of natural numbers to express natural number sequences, and as an added bonus, the difference between our bounds is equal to the length of the sequence.
Note that this part of the argument doesn't touch on 0 vs 1 yet, that is argued later in the essay.