http://img212.imageshack.us/img212/3649/84677187li3.gif i thought that lim inf is the smallest inf ever so why in this drawing its located on the right of inf(Xn) and they say that lim inf >= of point d how could i t be the subsequence converges to d .lim inf cannot be larger then D (it is D) and the same thing goes for the lim sup ??
If have no idea what "smallest inf ever" could mean! I suspect you have a completely wrong idea of what lim inf and lim sup are. "lim inf" of a sequence is the inf of all subsequential limits. "lim sup" of a seque nce is the sup of all subsequential limits Since I can't see the drawing you are talking about, I have no idea. If a sequence converges to anything, the all subsequences converge to the same thing so both lim inf and lim sup must be equal to that. For example, the sequence (-1)^{n}/n converges to 0 so both lim sup and lim inf are 0. That has nothing to do with the fact that there are members of the sequence both above and below 0. If the sequence were a_{n}= (-1)^{n} (n+1)/n, then there are two convergent subsequences: a_{n} with n even is (n+1)/n which converges to 1 and with n odd is -(n+1)/n which converges to -1. lim inf= -1 and lim sup= 1. However, all a_{n} with n even are greater than 1 and decrease to 1, while all a_{n} with n odd are less than -1 and increase to -1.