${1,2,3,4,5,6,7,8,9,10}$
sum of first 10 natural numbers = $\frac{10 . 11}{2}$ = 55
we need to show that there exist some combination of 3 numbers so that sum of those 3 numbers = 17
let us assume {1,a2,a3,a4,a5,a6,a7,a8,a9,a10}
sum of a2 + a3+........+ a9 = 55-1 = 54
if we choose any 3 numbers from a2 to a9 and try to evenly distribute the sum 54 between them,
$\frac{54}{3} = 18$
this shows that how evenly we pick numbers from 2 to 10 it can never go below 18 for some combination of a2 to a9.
18 is the optimal lower bound.