Seq Limits: Xn, Yn & Lim n→∞ = 0?

[dannysaf]

Let lim _{n →∞} X_nY_n = 0. Is it true that Lim_{n →∞} X_n= 0
or Lim_{n →∞}Y_n = 0 (or both)?

 

[njama]

lim _{n →∞} X_nY_n = lim _{n →∞} X_n * lim _{n →∞} Y_n

One or even both limits need to be 0 so that lim _{n →∞} X_n * Y_n =0

I guess you are right.

 

[VietDao29]

lim _{n →∞} X_nY_n = lim _{n →∞} X_n * lim _{n →∞} Y_n

Well, this is only true when both x_n, and y_n have limits. So, what if they don't? Say, what if they're oscillating?

 

[slider142]

lim _{n →∞} X_nY_n = lim _{n →∞} X_n * lim _{n →∞} Y_n

This equation is only correct if the individual limits on the right-hand side exist. The proposition in the original post is false and can be disproven with a counterexample.

 

[njama]

Yup, sorry I forgot to mention that the series must converge, in other way they would be divergent and we could not separate them.

 

[fmam3]

Simple counterexample. Clearly, $$\lim 1/n = \lim \(n \cdot 1/n^2 \) = 0$$ but we have that $$\lim n = +\infty$$ and $$\lim 1/n^2 = 0$$.

 

[Elucidus]

Well, this is only true when both x_n, and y_n have limits. So, what if they don't? Say, what if they're oscillating?

This example satisfies the original claim that at least one of the sequences converges to 0. A counter example for this must consist of two sequences whose product vanishes but both sequence do not converge to 0.

--Elucidus

 

[VietDao29]

This example satisfies the original claim that at least one of the sequences converges to 0. A counter example for this must consist of two sequences whose product vanishes but both sequence do not converge to 0.

--Elucidus

You've misquoted yet again, Elucidus..

You shouldn't forget to wear your glasses, then..

 

[Elucidus]

You've misquoted yet again, Elucidus..

You shouldn't forget to wear your glasses, then..

Indeed. I'm surprised. I was trying to quote this post:

Simple counterexample. Clearly, $$\lim 1/n = \lim \(n \cdot 1/n^2 \) = 0$$ but we have that $$\lim n = +\infty$$ and $$\lim 1/n^2 = 0$$.

To which my original comment makes more sense. Sorry for any confusion.

--Elucidus

 

[g_edgar]

Let $x_n$ be alternately 1 and 0, let $y_n$ be alternately 0 and 1.
Then $x_n y_n = 0$ for all n, but neither individual limit exists. In particular, neither individual limit is zero.

 

[HallsofIvy]

Let $x_n$ be alternately 1 and 0, let $y_n$ be alternately 0 and 1.
Then $x_n y_n = 0$ for all n, but neither individual limit exists. In particular, neither individual limit is zero.

Excellent example. However, neither $x_n$ nor $y_n$ converges and the O.P. finally told us.

Yup, sorry I forgot to mention that the series must converge, in other way they would be divergent and we could not separate them.

 

[njama]

Excellent example. However, neither $x_n$ nor $y_n$ converges and the O.P. finally told us.

HallsofIvy I am not the O.P, mate. I was just adding, additional explanation of my first post.