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I know it's quite obvious that $\limsup(a\cdot a_n)=a\cdot \limsup(a_n)$ for $a$ a real number >0, but I don't know how to prove it.

My second question is whether the following proof works for: $$\limsup(a + b) \leq \limsup(a) + \limsup(b)$$ for a and b as sequences.

http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2001;task=show_msg;msg=0119.0001.0001 Thanks!

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    It is false for $a<0$. Then $\limsup$ turns into $\liminf$.2012-12-02
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    For some basic information about writing math at this site see e.g. [here](http://meta.math.stackexchange.com/questions/5020/), [here](http://meta.stackexchange.com/a/70559/155238), [here](http://meta.math.stackexchange.com/questions/1773/) and [here](http://math.stackexchange.com/editing-help#latex).2012-12-02
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    What is $a,b$ in your second question?2012-12-02
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    A hint (assuming $a>0$): Do it for sup, then for the limits (which should be known already), then put the two together.2012-12-02
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    I am struggling to understand why an edit was reviewed and approved that made the question grammatically worse. Decapitalizing "I", replacing a period with ellipsis, and allowing the phrase "as an real number" are obviously wrong.2012-12-02
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    @rschwieb I agree and also don't understand why the link was deleted which at least at my PC seems to work.2012-12-02
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    Dear Promtea, show us what you have done and the community here will help you. I downvoted because the question is unclear (what are $a,b$? sequences, real numbers???) and shows no effort2012-12-02
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    @amWhy (and other people involved in the editing process): You can (and should) use `\limsup` rather than `\lim\sup` or `\mbox{limsup}`.2012-12-02
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    It is my first time posting on math.stackexchange, so I'm sorry I'm struggling with different functionalities.2012-12-02
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    For the inequality, you can look at these questions: [How to prove these inequalities in real analysis?](http://math.stackexchange.com/questions/205346/how-to-prove-these-inequalities-in-real-analysis), [Properties of $\liminf$ and $\limsup$ of sum of sequences](http://math.stackexchange.com/questions/70478/properties-of-liminf-and-limsup-of-sum-of-sequences) or [Subadditivity of the limit superior](http://math.stackexchange.com/questions/69391/subadditivity-of-the-limit-superior).2012-12-04

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