Homework Help: Existence of a natural number X

1. Nov 6, 2011

h.shin

1. The problem statement, all variables and given/known data
Given $M \in N$, show that there exists an $X \in N$ such that for all $n \geq X$, $n^2+n+1 \succ M$

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 $\geq$ X^2+X+1 since n $\geq$ X.
And X^2+X+1=M^2+M+1$\succ$M.
So, n^2+n+1$\succ$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?

2. Nov 6, 2011

LCKurtz

If that is the correct statement of the problem, I don't see anything wrong with your argument. But I have an uneasy feeling like you do. Since it seems so trivial I wonder if the original problem is mis-copied or misunderstood.