The proof uses the combinatorial "theorem", the Pigeonhole principle. This provides that S/134 is a good lower bound (dealing with all the possible orderings at the same time). As it is not an integer, but the answer to the question is, we actually got that the answer is at least the lowest integer not smaller than S/134. That's why the ceiling function is used.

Your proof doesn't give the exact example, but if an example of 15803 is provided, then you can see that this can actually be reached. So combining the two steps: you saw an example for a sum of 15803, but also got that there is no ordering for 15802 (or less).

The proof uses the combinatorial "theorem", the Pigeonhole principle. This provides that S/134 is a good lower bound (dealing with all the possible orderings at the same time). As it is not an integer, but the answer to the question is, we actually got that the answer is at least the lowest integer not smaller than S/134. That's why the ceiling function is used.

Your proof doesn't give the exact example, but if an example of 15803 is provided, then you can see that this can actually be reached. So combining the two steps: you saw an example for a sum of 15803, but also got that there is no ordering for 15802 (or less). You can look at the example as the step providing the upper bound for the proof for the correctedness of the answer.