am i correct with bolzano Weierstrass lim inf/sup definition..

transgalactic

i am given a bounded function
[itex]
y_n
[/itex]
by the law of bolzano Weierstrass
there has to be a sub sequence which converges.
[itex]
y_n_k
[/itex]

why ??
[itex]
lim sup y_n=>lim y_n_k
n-> infinity
[/itex]

lim sup is the supremum of the limits group of Xn
supremum is the least upper bound.so its bigger then any limit of a subsequence

correct?