Can You Visualize the Definition of Lim Inf Using a Simple Example?

[transgalactic]

my question and where i got stuck in the solution are in this link:

http://img244.imageshack.us/img244/5316/84661104oi7.gif

 

[HallsofIvy]

You have two sequences {x_n} and {y_n} and say that "by Bolzano-Weierstrasse", $\underline{\lim_{n\rightarrrow \infty}x_n}\le \lim_{n\rightarrow\infty}x_{N_n}$.

I will guess, from your other posts, that the underline is meant to be "lim inf" but what is $x_{N_n}$? An arbitrary subsequence of the sequence? If so, that would appear to follow directly from the definition of "lim inf" as the infimum of the set of all subsequential limits.

You then refer to the "bounded series y= sin(x)" which is neither a series nor a sequence but a function.

 

[transgalactic]

so the lim inf is the infimum of the limits of all the sub sequences

can you give a visual example that shows that

lim inf Xn<=lim Xn by that lim inf definition

i can't link them together

??