Remark Since any tail of a convergent sequence has the same limit, the hypotheses of theorems 3.2.4, 3.2.5, 3.2.6, and 3.2.7 can be weakened to apply to the tail of a sequence For example, in Theorem 3.2.4, if X = (x_n) is “ultimately positive” in the sense that there exists all m € N such that x_n ≥ 0 for all n ≥ m, then the same conclusion that x ≥ 0 will hold. Similar modifications are valid for the other theorems, as the reader should verify.