I need to prove that the following sequence converges:
$\lim_{n\rightarrow \infty} \frac{2n^2+3n+1}{n^2+n+1}=2$
So for the proof/solution I have the following:
Let $\epsilon >0$. Then let $N=\frac{1}{\epsilon}$. Then for all $n\geq N$, $|\frac{2n^2+3n+1}{n^2+n+1}-2| = |\frac{n-1}{n^2+n+1}| < \frac{1}{n} < \frac{1}{N} = \epsilon$
Thus the sequence converges to 2.
Is this the correct way of going about this? Thanks in advance.