How would you prove that this sequence converges to $\frac54$? How would you prove that this sequence converges to $\dfrac 54$? $$a_{n}=\dfrac {5n}{4n-3}$$
Thanks.

1) Intuitive  proof: divide numerator and denominator through $n$ and take the limit. Assume $\frac{1}{n} \to_n 0$.
2) Rigorous: consider $|a_n - \frac{5}{4}| < \varepsilon$. By the definition of convergence, we must find $n_0$ s.t. all $  a_{n \geq n_0}$ are within $\varepsilon-$radius of the limit. Take $n_0 = \frac{15+12 \varepsilon}{16 \varepsilon}$. Since $\varepsilon$ is arbitrary, the convergence is proven.
