How to use epsilon K proof to show a limit using Calculus?

[Winzer]

Homework Statement

Use $$\epsilon$$ K proof to show:
$$lim \left(\frac{n^2 + 2n + 1}{2n^2 + 3n + 2}\right) = \frac{1}{2}$$

Homework Equations

Hint first show
$$\left| \frac{n^2 + 2n + 1}{2n^2 + 3n + 2}-\frac{1}{2}\right| \leq \frac{1}{2n}, \hspace{0.5cm} n\epsilon N$$

The Attempt at a Solution

See the pdf. Please let me know if my argument can be more thorough.

 

[HallsofIvy]

At one point you have
$$\frac{1}{n(2n+1)}< \epsilon \doublearrow \frac{1}{2n}< \epsilon$$
What you should say is "Because
$$\frac{1}{2(2n+1)}< \frac{1}{2n}$$
if
$$\frac{1}{2n}< \epsilon$$
it will be true that
$$\frac{1}{2(2n+1)}< \epsilon$$