Example of an infinite sequence of irrational numbers converging to a rational number?

Pick any irrational number $\alpha$ you like, then consider the sequence $\{x_n=\alpha/n\}_{n=1}^\infty$. Then each term of the sequence $x_n$ is irrational and it converges to zero as $n$ tends to infinity.


How about the sequence $\sqrt{2}/n$, converging to zero?

Or, if you want to see some pattern with the decimal expansion, how about $\frac{\sqrt{2}}{10^n}$ giving 0.141421..., 0.0141421..., 0.00141421... ?


$\large\sqrt[n]{n}$ is irrational when $n>1$, and $\lim\limits_{n\to\infty}\large\sqrt[n]{n}=1$.

See also the related question Existence of irrationals in arbitrary intervals.

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Sequences And Series

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