Answer to Question #151781 in Calculus for Cypress

Question

Using the ε-N definition of convergence of sequences, prove that limit of (2/n) when n goes to infinity is 0.

Answer 1

We begin by examining the size of the difference:

$$\big|\dfrac{2}{n}-0\big|=\dfrac{2}{n}$$

Given $\varepsilon>0$

$$\big|\dfrac{2}{n}-0\big|=\dfrac{2}{n}<\varepsilon, \text{if}\ n>\dfrac{2}{\varepsilon}$$

Therefore using the limit definition

$$\lim\limits_{n\to \infin}(\dfrac{2}{n})=0$$

The sequence converges to  when $n$ goes to infinity.