prove that [x] + 1 = [x + 1]

Question

prove that [x] + 1 = [x + 1]

Answer 1

[x] + 1 = y, y an integer iff

[x] = y- 1, y an integer, iff

y - 1 <= x < y, y an integer, iff

y - 1 + 1 <= x + 1 < y + 1, y an integer, iff

y <= x + 1 < y + 1, y an integer, iff

[x + 1] = y, y an integer.

[x] + 1 = y, y an integer iff [x + 1] = y, y an integer.

Thus, [x] + 1 = [x + 1]