Bolzano-Weierstrass Theorem: Let S C R^n. Then S is compact (bounded and closed) iff every sequence of points in S has a convergent subsequence whose limit lies in S. I am completely completely lost when reading this example. 1. Why do we need 2 cases? 2. How are the 2 cases different? 3. For the second case, how come the subscripts of x_n_j and L_i_j are different? (n and i) 4. I don't understand the solution at all, can somebody please explain it step-by-step what is happening? I really want to understand this example! Thanks a lot!