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## Homework Statement

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]

## Homework Equations

## 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?