To prove that the sequence $\{a_n\} = \left\{\frac{3n - 1}{4n + 5}\right\}$ is convergent, we need to find its limit as $n \to \infty$.

Step-by-step proof:

1. Expression of the sequence: $a_n = \frac{3n - 1}{4n + 5}$

2. Divide numerator and denominator by $n$:

   To simplify the expression as $n$ approaches infinity, divide both the numerator and the denominator by $n$:

   $a_n = \frac{\frac{3n - 1}{n}}{\frac{4n + 5}{n}} = \frac{\frac{3n}{n} - \frac{1}{n}}{\frac{4n}{n} + \frac{5}{n}} = \frac{3 - \frac{1}{n}}{4 + \frac{5}{n}}$

3. Take the limit as $n \to \infty$:

   Now, we can take the limit of the simplified expression as $n$ approaches infinity:

   $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{3 - \frac{1}{n}}{4 + \frac{5}{n}}$

   As $n \to \infty$, $\frac{1}{n} \to 0$ and $\frac{5}{n} \to 0$. So, the expression simplifies to:

   $\lim_{n \to \infty} a_n = \frac{3 - 0}{4 + 0} = \frac{3}{4}$

4. **Conclusion