1. The problem statement, all variables and given/known data Given [itex]M \in N[/itex], show that there exists an [itex] X \in N [/itex] such that for all [itex] n \geq X [/itex], [itex] n^2+n+1 \succ M [/itex] 2. Relevant equations 3. The attempt at a solution Since both M and X are natural numbers and I am just trying to prove the existence of a certain natural number X, I thought that i could just set X = M. Then, n^2+n+1 [itex]\geq[/itex] X^2+X+1 since n [itex]\geq[/itex] X. And X^2+X+1=M^2+M+1[itex]\succ[/itex]M. So, n^2+n+1[itex]\succ[/itex]M. Is this a sufficient proof for the existence of X? It just doesn't feel like a full proof, should X be more limited?