Question about using arbitrary $\epsilon$ in real analysis proofs I've notice that in a lot of the proofs that are assigned in an undergraduate analysis course, we are often trying to show that some quantity is bounded by an arbitrary epsilon. 
For example, if I want to show that 
$$\lim_{n\to\infty}\frac{2}{\sqrt{n}}+\frac{1}{n}+3 = 3$$ 
I could try to show that for any $\epsilon > 0$ I can find an index $N$ such that 
$$\left| \frac{2}{\sqrt{n}}+\frac{1}{n} \right| < \epsilon$$
I would try to do something like using the Archimedean principle to show that if $\epsilon = \epsilon'^2/9$ (where $\epsilon'$ is dependent on the choice of $\epsilon$) that I can get find an index such that...
My real question is that how would I justify that any arbitrary $\epsilon$ can be written as $\epsilon = \epsilon'^2/9$? How do I justify that this equality would always have a solution? Would I appeal to density of the real numbers somehow?
Thanks
P.S. I think some of the answers are based on helping me prove the sequence converges, but I was mainly talking about how given some $\epsilon >0$, how I can assert that I can always find another real number $k$ such that $\epsilon = k^2/9$. Writing it this way makes the proof easier to read in my opinion, but I don't want to rely on something that seems obvious but I can't rigidly justify.

A certain tolerance  $\epsilon>0$ is  given to us. We have to find an $n_0$ such that 
$$0<r_n:=\left|{2\over\sqrt{n}}+{1\over n}\right|<\epsilon$$
for all $n\geq n_0$, but we are not required to find the smallest $n_0$ that does the job.
You already have remarked that $$0<r_n\leq {3\over\sqrt{n}}\qquad(n\geq1)\ .$$
If $n$ is large enough to make $${3\over\sqrt{n}}<\epsilon\tag{1}$$ then $r_n<\epsilon$ is guaranteed as well. Now $(1)$ is equivalent with
$$n>{9\over\epsilon^2}\ ,$$
and this is fulfilled for all $n$ with
$$n\geq n_0:=\left\lfloor{9\over\epsilon^2}\right\rfloor+1\ .$$
Note that we didn't have to construct any new real $\epsilon'$ or similar, but we implicitly have used the Archimedean principle in the definition of $n_0$.
