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A result from introductory analysis shows that given two bounded sequences $\{x_n\}$ and $\{y_n\}$,

$\liminf_{n\rightarrow\infty}(x_n+y_n)\geq\liminf_{n\rightarrow\infty}x_n+\liminf_{n\rightarrow\infty}y_n$.

I need to find at least two examples of where the above inequality is strict. One is if $x_n=(-1)^n$ and $y_n=(-1)^{n-1}$. Can anyone think of any other examples? I need at least one more.

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    $\begingroup$ $x_n = \sin(n)$ and $y_n = \cos(n)$? $\endgroup$
    – M.B.
    Apr 17, 2014 at 1:23
  • $\begingroup$ I don't think those sequences have any cluster points. If you meant $\{\sin(\frac{\pi}{2}-n\pi)\}$ and $\{\cos(n\pi)\}$, they're the same sequences, just written differently. $\endgroup$
    – mjh
    Apr 17, 2014 at 1:26
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    $\begingroup$ @mjh: no, that sequence is good: the smallest that $\sin(n)+\cos(n)=\sqrt2\sin(n+\pi/4)$ can be is $-\sqrt2$, yet each alone can reach $-1$. $\endgroup$
    – robjohn
    Apr 17, 2014 at 1:39

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I might as well give my comment in an answer. Try the following:

$$x_n = \sin(n)\qquad y_n = \cos(n)$$

For all $x$ we have that $\sin(x) + \cos(x) \geq -\sqrt{2}$, and $\sin(x) \geq -1$, $\cos(x) \geq -1$.

See: http://en.wikipedia.org/wiki/Equidistribution_theorem

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  • $\begingroup$ Yes, but what are the limit inferiors of $\{x_n\}$, $\{y_n\}$, and $\{x_n+y_n\}$? $\endgroup$
    – mjh
    Apr 17, 2014 at 1:31
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    $\begingroup$ $-1, -1, -\sqrt{2}$. $\endgroup$
    – M.B.
    Apr 17, 2014 at 1:34
  • $\begingroup$ The limit inferior of a bounded sequence with cluster points is the smallest cluster point of the sequence. What you're saying is that the sequence $\{-1\}$ is a subsequence of both $\{\sin(n)\}$ and $\{\cos(n)\}$. But $\sin(n)=-1$ or $\cos(n)=-1$ only if $n$ involves $\pi$, and $n$ is a natural number. $\endgroup$
    – mjh
    Apr 17, 2014 at 1:40
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    $\begingroup$ @mjh: $\sin(n)+\cos(n)=\sqrt2\sin(n+\pi/4)$ $\endgroup$
    – robjohn
    Apr 17, 2014 at 1:40
  • $\begingroup$ @mjh: sure but, $1,2,3,\ldots \pmod \pi$ comes arbitrarily close $\endgroup$
    – M.B.
    Apr 17, 2014 at 1:46

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