The discussion centers on the properties of limits of irrational sequences, specifically whether the limit of an irrational sequence, denoted as ##a{_n}##, is also irrational when the limit approaches a finite number ##b##. The participants explore the implications of induction in this context and inquire about existing theorems that might support or refute this claim. A specific example provided is the sequence defined by ##\frac{\pi}{n}+1##, which is irrational for all natural numbers ##n## not equal to zero.
PREREQUISITESMathematicians, students studying real analysis, and anyone interested in the properties of irrational numbers and sequences.
Hint ##\frac{\pi}{n}+1## is irrational (for ##n \in \mathbb N, n\neq 0##).Physicist97 said: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!