# Existence of a natural number X

## Homework Statement

Given $M \in N$, show that there exists an $X \in N$ such that for all $n \geq X$, $n^2+n+1 \succ M$

## 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?