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.

  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ @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

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 $\lim \inf b_n.$

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.