Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 7 Exercise 7.4

Solution:

### Part a

Let $x_n=\dfrac{1}{\sqrt{n^2+1}}$, then it is clear that all $x_n$ are irrational. However, the sequence converges to zero which is rational.

### Part b

We can always approximate an irrational number using rational numbers. For example, we know that

$$

\sqrt 2=1.4142135623\dots

$$ If we take a sequence $(x_n)$ as follows, $x_1=1$, $x_2=1.4$, $x_3=1.41$, $x_4=1.414$, $x_5=1.4142$, $x_6=1.41421$, and so on. Clearly, the sequence consists of rational numbers. However, the limit is $\sqrt 2$ which is irrational.