real analysis - Prove $lim_{a to infty} frac{3a+2}{5a+4}= frac35$ using definition of limit

Prove the limit using definition of limit

$$\ \lim_{a \to \infty}\ \frac{3a+2}{5a+4}= \frac{3}{5}$$

Answer: Let $\varepsilon \ >0$. We want to obtain the inequality

$$\left|\frac{3a+2}{5a+4}- \frac{3}{5}\right|< {\varepsilon}$$
$$\Rightarrow \left|\frac{3a+2}{5a+4} - \frac{3}{5}\right|\ =\left|\frac{5(3a+2)-3(5a+4)}{5(5a+4)}\right|\\= \left|\frac{-2}{5(5a+4)}\right|\le\frac{1}{a}$$
Therefore we choose $K \in N$ s.t $K> \frac{1}{\varepsilon}$
$$\Rightarrow\left|\frac{3a+2}{5a+4}- \frac{3}{5}\right| \le\frac{1}{a}\le\frac{1}{K} < \varepsilon$$ Is this correct?