Timeline for $x_n \to x \in \mathbb{R}$ and $y_n \to 0 $. What about $\frac{x_n}{y_n}$?

9 events

when toggle format	what		by	license	comment
S Jul 15, 2018 at 8:55	history	suggested	Noa Even	CC BY-SA 4.0	improved formatting
Jul 15, 2018 at 8:41	review	Suggested edits
S Jul 15, 2018 at 8:55
Jul 13, 2018 at 20:23	vote	accept	Student
Jul 13, 2018 at 20:14	answer	added	mechanodroid		timeline score: 1
Jul 13, 2018 at 20:05	history	edited	Student	CC BY-SA 4.0	added 320 characters in body
Jul 13, 2018 at 20:05	comment	added	user574380		What I said is such a proof, because you will use your argument above, which is one.
Jul 13, 2018 at 20:03	comment	added	Student		Thanks, but I am looking for a 'classic proof', using $n_0-\epsilon$. This question is one after basic results on convergent sequences (sum, difference, quotient rules etc.)
Jul 13, 2018 at 20:00	comment	added	user574380		You can always take bijections $a:\mathbb{N}\to y^{-1}((0,\infty))$ and $b:\mathbb{N}\to y^{-1}((-\infty,0))$ and apply your argument above (which is correct) to $x_{a_n}/y_{a_n}$ and to $x_{b_n}/y_{b_n}$. On the other hand, the proof cannot be the analysis of the single example that you have at the end. You want to prove the general statement, that for every $y_n\to0$ without definite sign and $x_n\to x$, the limit of $x_n/y_n$ is not one of $\pm\infty$.
Jul 13, 2018 at 19:47	history	asked	Student	CC BY-SA 4.0