Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm trying to show: $\inf(x_k+y_k) \leq \inf(x_k)+\inf(y_k)$. So I did:

Let $r>0$, there is a $k_1$ such that $x_{k_1}\leq \inf(x_k)+r/2$ and there is a $k_2$ such that $y_{k_2}\leq \inf(y_k)+r/2$.

Then: $x_{k_1}+y_{k_2} \leq \inf(x_k)+\inf(y_k)+r$

I'd like to take the inf and make $r \rightarrow 0$, but since $k_1$ and $k_2$ may be different I can't conclude. Please somebody help!

share|cite|improve this question
"Inf" is super-additive, while "Sup" is subadditive. – Siminore Sep 28 '12 at 11:17
up vote 1 down vote accepted

Let $x_k=(-1)^k$, $y_k=(-1)^{k+1}$. Then $x_k+y_k=0$ for all $k$, hence the $\inf$ is $0$ as well. However, $\inf x_k=\inf y_k=-1$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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