Since the sequence diverges, when you throw away the convergent subsequence (i.e. remove its elements from the sequence), you get another sequence. As it is defined on a compact set, this new sequence also has a limit point. Note that this new sequence need not be divergent, and so this process cannot necessarily be repeated indefinitely. For example, consider the sequence
$$1, -1, 1, -1, 1, -1, \cdots $$
in $[0,1]$.Throwing away the subsequence of $-1$'s gives us a convergent subsequence of $1$'s.
If the sequence converges, throwing away the convergent subsequence amounts to removing all the elements in the sequence, and so the result does not hold.