Pigeonhole Principle question

I can't seem to figure out how to construct the pigeonholes here.

Initially, as there are 999 natural numbers, I was thinking that I can split the this into 38 equal intervals which are $\approx 26.2895$ in length so taking the floor value of this would be 26 to go along with the notion of integers, but I'm not sure if this is the correct application of the principle?

I can't seem to figure out how to construct the pigeonholes here.

Initially, as there are 999 natural numbers, I was thinking that I can split the this into 38 equal intervals which are $\approx 26.2895$ in length so taking the floor value of this would be 26 to go along with the notion of integers, but I'm not sure if this is the correct application of the principle?

I'm not really sure why you're thinking in terms of intervals (you mean partitions or subsets) or why you want 38 of them. You want n partitions where n is one less than the number of choices you can make so that by the principle you can ensure two of your choices are in the same partition/subset. If you have 38 partitions then nothing is guaranteed, all 38 of your choices could be in 38 seperate partitions.

You have the natural numbers (I'm counting from 1) $\{1, \ldots, 999\}$ which you can partition into $\{1, \ldots, 27\}, \{28, \ldots, 53\}, \ldots, \{...,999\}$ of size 27 each. There are 37 of these subsets. So if you pick 38 natural numbers less than 1000 then two of them are...? Can you finish off from here?

Okay that seems much clearer than what I was going with before. So for 38 numbers it would make two of them be in the same partition and in maximum possible distance of 26 from each other. Would that be sufficient enough to say for the proof?