I am looking at the following problem: Given the finite set $\{2,3,4,5,6,7,8\}$ suppose we choose three distinct elements $a,b,c$ out of it. Clearly there are $\binom{7}{3}$ ways of doing that. Without going through all the cases I wish to prove that each case yields an $x,y\in\{a,b,c\}$ such that either $\{1,x,y\}$ contains no consecutive integers or $\{x,y,9\}$ contains no consecutive integers.
I have tried to cut down the cases by symmetric arguments, but can someone suggest a cleaner proof?
Thanks.