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## Homework Statement

Given a

**bounded**set x_n and for

**any y_n**the following condition holds:

[tex] \limsup_{n \rightarrow ∞}(x_n+y_n) = \limsup(x_n)+\limsup(y_n) [/tex]

Show that x_n converges.

## Homework Equations

Definition of limsup(x_n) = L:

[tex] \forall \epsilon > 0 \mid \exists N \mid \forall n > N \mid x_n < L +\epsilon [/tex]

## The Attempt at a Solution

I started by defining the limsups as follows:

Let limsup(y_n) = a; limsup(x_n) = b;

According to the given conditions limsup(x_n+y_n) = a + b(see above).

Therefore:

[tex]

\forall \epsilon > 0 \mid \exists N_1 \mid \forall n > N_1 \mid x_n+y_n < (a+b) + \epsilon \\

\mbox{Taking the same Epsilon:}\\

\exists N_2 \mid \forall n > N_2 \mid y_n < a + \epsilon \\

n>N \mid N = \max(N_1, N_2)

[/tex]

Now I arrive at the following congruent inequalities(to be subtracted by their transitive property):

[tex]

x_n+y_n < (a+b) + \epsilon \\

y_n < a + \epsilon \Rightarrow \\

x_n < b

[/tex]

Now, I claim that since I am given that x_n is bounded, by arriving at that final inequality, I've effectively discovered its

**upper bound**.

Therefore:

[tex]

x_n<b

[/tex]

But, since b is the limit superior, it is also the

**smallest possible, real upper bound**.

And yet, here I'm stuck.

On the one hand, the above relation means that -b is the largest

**possible lower bound**, or the limit inferior.

But that means that limsup(x_n)≠liminf(x_n) which would imply that x_n

**does not**converge.

Where should I turn next?

Is there perhaps a better way to look at this?

Very thankful for your attention,

Daniel