Prove if n<m there is at least one [(n/m)]?

Question

Suppose there are n programmers in m cubicles. Prove that there must be at least one cubicle containing at least $\lceil \frac{n}{m} \rceil$ programmers.

Note: I was not able to find the right sign [ is returning first upper integer in case of not integer number.

1.1 = 2
1.5 = 2
1.9 = 2

Answer 1

Prove by contradiction: suppose that no cubicles have at least $\frac{n}{m}$ programmers. Then the total amount of programmers is less than $n$, a contradiction. Also, the amount in each cubicle must be an integer, which proves the slightly stronger result with the ceiling function.