The problem I'm working on says:
A basketball player has been training for 112 hours during 12 days. He has trained an integer number of hours every day. Prove that there was two consecutive days where he has trained for at least 19 hours.
I'm following this rationale to try prove it:
As we are interested in two consecutive days we split the 12 days into 6 pairs: {1,2}, {3,4}, {5,6}, {7,8}, {9,10}, {11, 12}
Using a corollary of the Pigeonhole principle we know that the sum of one the pairs is greater than 18, but not 19.
Does this mean that I can't prove it this way?