 Many sequences that are not increasing have limits as well. For instance, the sequence \(1, -1/2, 1/3, -1/4, \dots, (-1)^n/n, \dots\) can be seen to converge to the value 0, even though it is not increasing. However, if a sequence is not bounded, like the sequence \(1, 2, 3, \dots\), it does not have a limit—if it had a limit, there would have to be some values \(l\), \(\epsilon\), and \(N\), such that all terms in the sequence beyond the first \(N\) are within \(\epsilon\) of \(l\). But then any number that is larger than the first \(N\) terms of the sequence, and also larger than \(l+\epsilon\), would be a bound for the sequence. And some sequences that are not increasing also fail to have a limit—for instance, the sequence \(1, 0, 1, 0, 1, 0, \dots\) does not have a limit, because there is no value such that all terms of the sequence are eventually within \(1/3\) of that value.