Show that for any k ≥ 2, if n + 1 distinct integers are chosen from...

Show that for any k ≥ 2, if n + 1 distinct integers are chosen from the set [kn] = {1, 2, . . . , kn}, then there will be two integers which differ by at most k − 1. Please demonstrate the steps so that I can learn from it and solve other problems by following the reasoning!

Solutions

Expert Solution

We first suppose that the given result is not true and will later see that this assumption leads us to a contradiction.

So, suppose we can choose n+1 distinct integers from the set [kn] = {1,2, . . . , kn} such that the difference between any two of them is at least k, Where .

Let these chosen integers are arranged in a increasing order and denote them as

So,

[ Since the difference between and is at least k]

This is a contradiction because cannot be larger than the largest element kn of the set [kn].

Hence our assumption leads us to a contradiction. So our assumption was wrong. Hence, for any , if n+1 distinct integers are chosen from the set [kn] = {1,2, . . . , kn} , then there will be two integers which differ by at most k-1.

(Proved)

Colby Messinger answered 1 year ago

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