1. ## lim inf/sup question...

i got this example:

http://img405.imageshack.us/img405/2299/64500275qt5.gif

i got a sequence of numbers
i cant understand from where they conclude that the limits of every sub sequence is {2,1,0,-1}

there are no sub sequences pointed out
there are no formulas for each sub sequence

how did they came with
that the limits of every sub sequence is {2,1,0,-1}
??

2. Originally Posted by transgalactic
i got this example:

http://img405.imageshack.us/img405/2299/64500275qt5.gif

i got a sequence of numbers
i cant understand from where they conclude that the limits of every sub sequence is {2,1,0,-1}

there are no sub sequences pointed out
there are no formulas for each sub sequence

how did they came with
that the limits of every sub sequence is {2,1,0,-1}
??
If $\displaystyle s_n$ is a convergent subsequence it means the the terms of this sequence have to get sufficiently close to eachother far enough into the sequence (this is Cauchy's criterion). The terms of the full sequence consist of only $\displaystyle -1,0,1,2$ it means if $\displaystyle s_n$ is a convergent subsequence then it must be eventually constant. Therefore, the only possible subsequential limits are $\displaystyle \{ -1,0,1,2\}$. The limsup is the largest of these subsequential limits and the liminf is the smallest of these subsequential limits.