The method of induction is typically used to prove statements about integers within mathematical proofs, but we can attempt an analogy to argue for the similarity of the night sky over distances like 1,900 miles.

Step 1: Define the Claim

We aim to show that two observers separated by 1,900 miles see essentially the same night sky. This means the arrangement of stars and celestial objects appears nearly identical.

Step 2: Base Case

Consider two observers who are standing very close together—say, just a mile apart. At such a small separation, the difference in their viewing angles of celestial objects is negligible, meaning they see essentially the same night sky.

Consider that you can repeat this step for a third person, compared with the second person, and then compare the night sky for the third person and the first, and so on.

Step 3: Inductive Step

Assume that for a given separation distance d, two observers see nearly the same night sky. We now show that an observer moving an additional small distance \Delta d (e.g., a mile farther) still sees nearly the same sky.

Since the stars are incredibly far away—on the order of light-years—the difference in viewing angle due to moving a mile (or even dozens of miles) is minuscule. The shift in perspective for each star, given the vast distances involved, is negligible. Therefore, if the night sky is essentially the same for an observer at d miles, it will also be the same for an observer at d + \Delta d miles.

By repeating this argument iteratively (inductively increasing the separation distance in small increments), we extend it to any large distance, including 1,900 miles.

Step 4: Conclusion

By induction, the night sky remains nearly identical for observers as we increase their separation gradually, even up to 1,900 miles. This is because the stars are so far away that their apparent positions do not change significantly over such distances on Earth.

J/k, it’s totally different unless you are both on the same rotational plane.