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Sequence of rational numbers has an irrational limit


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.


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