Rationality of a Sequence

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!
 

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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##).
 

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