Show that in the sequence: x, 2x, 3x, ....
analysis/sequence
Problem
Show that in the sequence: x, 2x, 3x, .... (n-1)x (x can be any real number) there is at least one number which is within 1/n of an integer.
Show solution
Throw 0 into the sequence; there are now n numbers, so some pair must have fractional parts within 1/n of each other; their difference is then within 1/n of an integer.