# Why why why (tiny limsup proof)

1. ### quasar987

4,770
There's a "theorem" in my book that says if x is a cluster point of {x_n}, then lim inf(x_n)$\leq$x$\leq$lim sup(x_n).

The way the author proves it is a little bit extravagant. Why not just say "Let A be the set of all cluster points. Then, lim inf(x_n)=inf(A)$\leq$x$\leq$sup(A)=lim sup(x_n)."

???

2. ### Dick

25,735
Don't they pretty much say the same thing?