By definition, a sequence of real numbers is a function where we denote each term . So, every sequence has infinitely many terms (although a term may be repeated infinitely many times). Since is bounded, it is entirely contained in the interval . Splitting the interval in half, we see that the sequence is entirely contained within , since this union is the same as the original interval. Now, since the sequence has infinitely many terms (though not necessarily infinitely many distinct terms), one of those two halves of [a,b] has to contains for infinitely many of the ns, otherwise there would be terms outside the interval [a,b] (which we're assuming is not the case).
Does that help?
Edit: It doesn't say that the sub-interval contains every term of the sequence. It just says that it contains an infinite number of the terms.