$\epsilon - N$ proof of $\sqrt{4n^2+n} - 2n \rightarrow \frac{1}{4}$

Question

I have the following proof for $\lim_{n\rightarrow\infty} \sqrt{4n^2+n} - 2n = \frac{1}{4}$ and was wondering if it was correct. Note that $\sqrt{4n^2+n} - 2n = \frac{n}{\sqrt{4n^2+n} + 2n}$. $$\left|\frac{n}{\sqrt{4n^2+n} + 2n} - \frac{1}{4}\right| \\ = \left|\frac{2n - \sqrt{4n^2+n}}{4(\sqrt{4n^2+n} + 2n)}\right|=\left|\frac{\sqrt{4n^2+n} - 2n}{4(\sqrt{4n^2+n} + 2n)}\right|\\ = \left|\frac{n}{4(\sqrt{4n^2+n} + 2n)^2}\right| \leq \left|\frac{n}{4(4n)^2}\right| = \left|\frac{n}{64n^2}\right| \\ = \left|\frac{1}{64n}\right| < \epsilon \\ \implies n>\frac{1}{64\epsilon}$$

Answer 1

You want to show that $\lim_{n\rightarrow\infty} \sqrt{4n^2+n} - 2n = \frac{1}{4}$. To do this, I would split up the analysis into scratch work and the formal proof.

For the scratch work, you need to find a suitable upper bound. You have done this by showing

\begin{align} \left|\frac{n}{\sqrt{4n^2+n} + 2n} - \frac{1}{4}\right| & = \left|\frac{2n - \sqrt{4n^2+n}}{4(\sqrt{4n^2+n} + 2n)}\right|\\&=\left|\frac{\sqrt{4n^2+n} - 2n}{4(\sqrt{4n^2+n} + 2n)}\right|\\& = \left|\frac{n}{4(\sqrt{4n^2+n} + 2n)^2}\right| \\&\leq \left|\frac{n}{4(4n)^2}\right| \\&= \left|\frac{n}{64n^2}\right| \\& = \left|\frac{1}{64n}\right| \\&< \epsilon \end{align}

which means that $n>\frac{1}{64\epsilon}$ is the upper bound.

For the formal proof:

Let $\epsilon>0$ (you need to fix $\epsilon$ as a small positive constant). It follows from $\frac{1}{\epsilon}>0$ that $\frac{1}{64\epsilon}>0$. Then by the Archimedean property there exists a $N\in\mathbb N$ such that $N>\frac{1}{64\epsilon}$. Then if $n\geq N > \frac{1}{64\epsilon}$, we have

$$\left|\frac{n}{\sqrt{4n^2+n} + 2n} - \frac{1}{4}\right|\leq \frac{1}{64n}<\epsilon $$

where the last inequality follows from

$$n\geq N > \frac{1}{64\epsilon} \implies n>\frac{1}{64\epsilon} \implies\epsilon > \frac{1}{64n}$$

Answer 2

Like the other commenters have mentioned, the implication is in the wrong direction. Since you want $|\frac{1}{64n}| < \epsilon$, you are required to have $n > \frac{1}{64 \epsilon}$. In other words, $n > \frac{1}{64 \epsilon}$ implies $|\frac{1}{64n}| < \epsilon$. What you have written is the other way around.

Answer 3

You did the scratch work correctly. But I wouldn't call this a proof (of course it contains all ingredients of a good proof!).

You didn't introduce $\epsilon$. Of course, everyone knows what you mean but one should write it out if one wants to be fully rigorous.

So, your proof should read:

Let $\epsilon >0$. Let $N$ be an integer greater than --insert your choice for $N$ here--.

If $n \geq N$, we have

--insert your estimations here--

Thus, by definition of limit, the result follows. QED