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Let $\{a_n\}_{n=1}^\infty$ and $\{b_n\}_{n=1}^\infty$ be two bounded sequences (not necessarily convergent) of positive real numbers.

Is it true that $$\liminf_n \ a_n+\liminf_n \ b_n=\liminf_n \ (a_n+b_n)?$$

Any information/reference/comment will be appreciated.

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    $\begingroup$ Take $a_n=2+(-1)^n$ and $b_n=2+(-1)^{n+1}$. They are bounded and positive. What are their inferior limits, and the inferior limit of their sum? $\endgroup$ – user539887 Apr 23 '18 at 12:40
  • $\begingroup$ @user539887 That's a very good hint. Why don't you post it as an answer (saying that it is a hint)? $\endgroup$ – José Carlos Santos Apr 23 '18 at 12:44
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    $\begingroup$ @user539887: 1,1 and 4. Thanks $\endgroup$ – serenus Apr 23 '18 at 12:45
  • $\begingroup$ On the other hand, the inequality that you see is true in general: Fatou's lemma $\endgroup$ – user551819 Apr 23 '18 at 12:46
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NO. Example: For $n\in \Bbb N$ let $a_{2n}=0$ and $a_{2n-1}=1$ and $b_n=a_{n+1}.$ Then $\lim \inf a_n=\lim \inf b_n=0$ but $a_n+b_n=1$ for every $n.$

Let $A_n= \inf_{m>n}a_m$ and $B_n=\inf_{m>n}b_m$ and $C_n=\inf_{m>n}(a_m+b_m).$ We have $A_n+B_n\leq C_n$ for every $n$, so $$\lim \inf a_n+\lim \inf b_n=\lim_{n\to \infty}A_n+\lim_{n\to \infty} B_n=\lim_{n\to \infty}(A_n+B_n)\leq$$ $$\leq \lim_{n\to \infty}C_n=\lim \inf (a_n+b_n) .$$

Note that $\lim \inf a_n$ is the least value that any convergent sub-sequence of $(a_n)_n$ can converge to. In order for $\lim \inf a_n +\lim \inf b_n=\lim \inf (a_n+b_n)$ it is necessary that there exists a strictly increasing $f:\Bbb N\to \Bbb N$ such that $(a_{f(n)})_n$ converges to $\lim \inf a_n$ and such that $(b_{f(n)})_n$ converges to $\lim \inf b_n.$

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  • $\begingroup$ Thanks for the example and the final remark which will be surprisingly useful for me. In fact, I had two sequences with the property given in the remark. $\endgroup$ – serenus Apr 24 '18 at 17:26
  • $\begingroup$ But, are the conditions (in your final remark ) sufficient to have $\liminf a_n +\liminf b_n =\liminf (a_n +b_n) $, too? $\endgroup$ – serenus Apr 24 '18 at 18:36

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