If the first term is positive and the last term is negative, then the sequence has to pass 0 (or go past infinity, in some cases, but we'll ignore that) somewhere on that interval.

Because the sequence must pass from one integer to the next higher or next lower integer, then it must pass through the integer zero.

I'm unfamiliar with the discrete analogue of the intermediate value theorem, so I can't help you there. However, the result here should be intuitive.