- #1

danielakkerma

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(Hello everyone!)

Given that [tex]\limsup_{n \rightarrow \infty}(\frac{1}{x_n})\cdot \limsup_{n \rightarrow \infty}(x_n)=1[/tex]

Show that x_n converges.

Recalling that:

[tex] x_n \text{ converges } \iff \liminf(x_n)=\limsup(x_n)[/tex]

Started with this:

Firstly, let's designate the limsup of (x_n) as L;

Therefore, according to the stipulated conditions, limsup(1/x_n) = 1/L;

Now:

[tex]1. \forall \epsilon >0 \; \exists N \mid \forall n>N \mid x_n \leq L+\epsilon[/tex]

However, for the limit inferior:

[tex]2. \forall \epsilon >0 \; \exists N^{*} \mid \forall n>N^{*} \mid x_n \geq L-\epsilon[/tex]

Here arises my first question: can I unite both Ns, by finding the *maximum* of both, i.e. N'=Max{N, N*}?(Even though, I am not sure that in fact, higher orders of x_n, would find themselves in the inferior limit region).

And if not, how do I proceed?

(I tried inverting ineq. 2), to arrive at a value for limsup(1/x_n), but, to no avail.

Any help will be much appreciated,

(And I'm sorry if I bungled some of the definitions here, it's been a while since I had last done this ,

Most beholden,

And grateful as always,

Daniel

## Homework Statement

Given that [tex]\limsup_{n \rightarrow \infty}(\frac{1}{x_n})\cdot \limsup_{n \rightarrow \infty}(x_n)=1[/tex]

Show that x_n converges.

## Homework Equations

Recalling that:

[tex] x_n \text{ converges } \iff \liminf(x_n)=\limsup(x_n)[/tex]

## The Attempt at a Solution

Started with this:

Firstly, let's designate the limsup of (x_n) as L;

Therefore, according to the stipulated conditions, limsup(1/x_n) = 1/L;

Now:

[tex]1. \forall \epsilon >0 \; \exists N \mid \forall n>N \mid x_n \leq L+\epsilon[/tex]

However, for the limit inferior:

[tex]2. \forall \epsilon >0 \; \exists N^{*} \mid \forall n>N^{*} \mid x_n \geq L-\epsilon[/tex]

Here arises my first question: can I unite both Ns, by finding the *maximum* of both, i.e. N'=Max{N, N*}?(Even though, I am not sure that in fact, higher orders of x_n, would find themselves in the inferior limit region).

And if not, how do I proceed?

(I tried inverting ineq. 2), to arrive at a value for limsup(1/x_n), but, to no avail.

Any help will be much appreciated,

(And I'm sorry if I bungled some of the definitions here, it's been a while since I had last done this ,

Most beholden,

And grateful as always,

Daniel

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