Let s be the smallest integer. Then the 5 numbers are: s, s+1, s+2, s+3, s+4
Their average is (5s + 10)/5 or s + 2
So: s + 4 < 2(s + 2)
s + 4 < 2s + 4
s >0
So the smallest integer greater than 0 is 1.
Edited to correct mistake. Thanks to Guest below for noting this.