# Find $\lim\limits{\inf(a_{n})}$, $\lim\limits{\sup(a_{n})}$, $\inf\{a_{n}\}$, ${\sup(a_{n})}$

Given
$$a_{n}=\frac{(-1)^{n}}{n}+\frac{1+(-1)^{n}}{2}$$

Compute $$\lim\limits{\inf(a_{n})}$$

$$\lim\limits{\sup(a_{n})}$$

$$\inf\{a_{n}\}$$

$${\sup(a_{n})}$$

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where's the sequence? –  Adam Rubinson Apr 26 '12 at 23:30
Excuse me because there was as it should be the sequence –  Daniela del Carmen Apr 26 '12 at 23:33
Daniela, note you can decompose $a_n$ into two subsequences, namely $$a_{2n}=\frac 1 n +1$$ $$a_{2n+1} = -\frac 1 n$$ Maybe that makes the analysis easier. Note the former is always positive and decreasing, and the latter is always negative and increasing. Also, the former is always greater than the latter. –  Pedro Tamaroff Apr 26 '12 at 23:51
I'm a bit confused on the notation. Do you mean $\limsup$ or $\lim\limits_{n \to \infty}\sup a_n$? As far as I know, $\sup a_n$ will be a constant, with no dependance on $n$. –  Pedro Tamaroff Apr 26 '12 at 23:56
Daniela: When you mention the book you could perhaps tell us name of the book, page, number of exercise. In my opinion it is always good to give as much background and details as possible. (E.g. it might be possible that some of the users has the book or this books is accessible via Google books, and if this is the case, it might happen that someone will be able to explain you the notation from the book or notice some important details you missed when posting the question.) –  Martin Sleziak Apr 27 '12 at 7:06

I will assume that you're asking about $\limsup$ and $\liminf$. (This is, in my opinion the only reasonable explanation.)
$a_{2k}=1+\frac1{2k}$
$a_{2k+1}=-\frac1{2k+1}$
Can you find limits of these subsequences? What does this tell you about $\limsup$ and $\liminf$. (I guess that you have already learned about relationship between convergent subsequences and limit inferior and limit superior. Maybe this was even in the definition of $\liminf$ and $\limsup$ - not every teacher uses exactly the same definition.)
From the above you should have some inequalities for the values of $\liminf a_n$ and $\limsup a_n$, if you are able to prove the opposite inequalities, you're done. So for this part: Are you able to show that for each $n$ you have $-\frac1n\le a_n \le 1+\frac1n$? What do these inequalities tell you about $\liminf a_n$ and $\limsup a_n$?