<s>[INST] Prove that the sequence converges to using the $\varepsilon$-$N$ definition of a limit Question: Prove that the sequence $\{a_n\}_{n=1}^\infty$ defined by $a_n=\frac{2n}{3n+1}$ converges to $\frac{2}{3}$ using the $\varepsilon$-$N$ definition of a limit.

I'm still a beginner when it comes to these types of questions. I've had a go at it but I'm pretty sure this is nonsense. Would appreciate some feedback and suggestions. Thanks!

Solution: Fix $\varepsilon>0$. We need to find $N\in\mathbb{N}$ such that $n>N \implies|a_n-\frac{2}{3}|<\varepsilon.$
We have $|a_n-\frac{2}{3}| < \varepsilon.$
$\iff |\frac{2n}{3n+1}-\frac{2}{3}| < \varepsilon$
$\iff |\frac{2n}{3n+1}-\frac{2}{3}| < \varepsilon$
Now $2n\ge2$ for all $n\ge1$ and $3n+1\ge3n$ for all $n$, so we have
$\iff |\frac{2}{3n}-\frac{2}{3}| < \varepsilon$
$\iff |\frac{2}{3}(\frac{1}{n}-1)| < \varepsilon$
The sequence is positive for all $n$ so the absolute values are redundant
$\iff \frac{2}{3}(\frac{1}{n}-1) < \varepsilon$
$\iff \frac{1}{n}-1 < \frac{3\varepsilon}{2}$
$\iff \frac{1}{n} < \frac{3\varepsilon}{2}+1$
$\iff \frac{1}{n} < \frac{3\varepsilon+2}{2}$
$\iff n> \frac{2}{3\varepsilon+2}$
So choose any $N>\frac{2}{3\varepsilon+2}$ and the definition is satisfied. Q.E.D.
 [/INST] HINT: it is $$\frac{2n}{3n+1}-\frac{2}{3}=\frac{-2}{3(3n+1)}$$
and $$\left|\frac{-2}{3(3n+1)}\right|=\frac{2}{3(3n+1)}$$
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