Understanding Limits of Subsequences: A Lim Inf/Sup Question

[transgalactic]

i got this example:

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

i got a sequence of numbers
i can't 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}
??

 

[HallsofIvy]

The obvious subsequences is {2, 2, 2, 2, ...}, {1, 1, 1, 1, ...}, {0, 0, 0, ...}, and {-1, -1, -1, 0, ...}. Those are the only convergent subsequences.