Is the Limit of an Irrational Sequence Also Irrational?

[Physicist97]

Hello! So let's say that you have a sequence $a{_n}$ and the limit as $n->{\infty}$ gives the finite number $b$ not equal to zero. If $a{_n}$ is known to be irrational, and $a{_n}{_+}{_1}$ can be shown to be irrational, does it follow by induction that $b$ is irrational? Is there any theorem that states something equivalent to this, or is this not true for all cases? Thank you!

 

[Samy_A]

Hello! So let's say that you have a sequence $a{_n}$ and the limit as $n->{\infty}$ gives the finite number $b$ not equal to zero. If $a{_n}$ is known to be irrational, and $a{_n}{_+}{_1}$ can be shown to be irrational, does it follow by induction that $b$ is irrational? Is there any theorem that states something equivalent to this, or is this not true for all cases? Thank you!

Hint $\frac{\pi}{n}+1$ is irrational (for $n \in \mathbb N, n\neq 0$).