proof of bolzano-weierstrass theorem

[Jongki Cho]

There exist bounded infinite set S in R^n, if and only if every subset
of S has s limit point in S.

that is the bolzano-weierstrass theorem.

but I can't proof this way.

assume unbounded set S in R^n. -> an infinite subset T of S cannot
have a limit point.

please answer this proof.

[Zdislav V. Kovarik]

If your textbook presents a standard theorem, such as
Bolzano-Weierstrass Theorem, but does not offer the proof,
get a better textbook.

And your wording does not come close to what is known as
Bolzano-Weierstrass Theorem. Just for the fun of it, show your
version to someone around you who is familiar both with your
first language and with English, and have him/her translate it
back. Then compare it with what the theorem really says.
(English is not my first language, either, so I know how hard it
is in the beginning.)

Cheers, ZVK(Slavek).