So I have been trying to prove the following question from my textbook:
Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be two real convergent sequences, with $\lim a_n = a$ and $\lim b_n = b \neq 0$. Prove that $\lim \frac{a_n}{b_n} = \frac{a}{b}$. Why doesn't the requirement "$b_n \neq 0$ for all n" play a big role here?
Now I was able to prove that the limit of the quotient sequence is indeed the quotient of the limits. However, I just can't quite understand why the requirement "$b_n \neq 0$ for all n" doesn't have a big role here. Isn't that requirement necessary for the existence of the quotient sequence?
Best Answer
Strictly speaking, yes. However, as $b\ne 0$, we can at least be sure that only finitely many terms need to be removed from the sequence. And the convergence behaviour (including its possible limit) does not change when we modify the sequence in only finitely many terms. In that light, it doesn't even really matter when some (but only finitely many!) terms are not even defined in the first place.