Lim Superior tt53

Limit Superior & Limit Inferior Suppose first that X = (xn ) is bounded above, say by M . Let sn = sup{ xk : k ≥ n }. (Using the language of tail sequences, sn = sup Xn .) Then (sn ) is non-increasing. (As sets, the tail sequences are nested; and the sup over a smaller set cannot be larger.) Either sn is unbounded (below), or sn is convergent. Definition Let X = (xn ) be a sequence of real numbers that is bounded above, and let sn = sup{ xk : k ≥ n }. (a) If (sn ) converges, we define lim sup xn to be lim sn . (b) If (sn ) diverges to minus infinity, we write lim sup xn = −∞. Definition If X = (xn ) is a sequence of real numbers that is not bounded above, we write lim sup xn = ∞. Theorem If X = (xn ) is a convergent sequence of real numbers, then lim sup xn = lim xn . Proof Let L = lim xn . For any > 0, there exists M ∈ N such that L − < xn < L +

(n ≥ M ).

Thus, if n ≥ M , L + is an upper bound for Xn and L − is not an upper bound. That is, L − < sup Xn ≤ L + . It follows that L − ≤ lim sup xn ≤ L + . Since is arbitrary, lim sup xn = L. We define the limit inferior analagously. Definition Let X = (xn ) be a sequence of real numbers that is bounded below, and let tn = inf{ xk : k ≥ n }. (a) If (tn ) converges, we define lim inf xn to be lim tn . (b) If (tn ) diverges to infinity, we write lim inf xn = ∞.

Definition If X = (xn ) is a sequence of real numbers that is not bounded below, we write lim inf xn = −∞. It should not be surprising that if lim xn = x, then lim inf = x and lim sup xn = x. In fact, the converse is also true: Theorem If (xn ) is a sequence of real numbers and if lim inf xn = lim sup xn = x, then lim xn = x. Proof Let > 0. By hypothesis we have x = lim inf xn = lim(inf Xn ). Thus there exists K1 such that | inf Xn − x| <

(n ≥ K1 ).

In particular, x − < xn

(n ≥ K1 ).

Similarly, since lim sup xn = x, there exists K2 such that xn < x +

(n ≥ K2 ).

Thus for n ≥ K = max{ K1 , K2 }, we have x − < xn < x + . This establishes that lim xn = x. Here is another way to view limit superior and limit inferior. Theorem Let (xn ) be a bounded sequence of real numbers. 1. If lim sup xn = s, then for any > 0, (a) xn < s + for all but a finite number of values of n; and (b) xn > s − for infinitely many values of n. 2. If lim inf xn = t, then for any > 0, (c) xn > t − for all but a finite number of values of n; and (d) xn < t + for infinitely many values of n. Proof We will prove 1. If (a) were false, then for some > 0, we would have xn ≥ s + for infinitely many values of n. But then, for any n, Xn (the tail sequence) would contain a number greater than or equal to s + . Therefore, lim sup xn = lim(sup Xn ) ≥ s + which contradicts our hypothesis. Now suppose (b) is false. Then for some > 0, xn ≤ s − for all but a finite number of values of n. But then there exists K ∈ N such that xn ≤ s −

(n ≥ K).

This implies lim sup xn ≤ s − .

→←

From this, it follows that any bounded sequence of real numbers has a sequence that converges to its limit superior. (Sketch. Let s = lim sup xn . There are infinitely many n such that xn > s − 1. Let n1 be one such n. There are infinitely many n such that xn > s − 1/2. Choose n2 > n1 to be one such n. Etc.) It should also be clear that if (xn ) has a subsequence converging to y, then y cannot be larger than s = lim sup xn . (Sketch. Suppose there is such a subsequence. Then there are infinitely many of the subsequence “close” to y. But only finitely many are more than a little bigger than s. [I.e., let = (y − s)/2.]) Combining, we have this characterization of limit superior. Theorem Let (xn ) be a bounded sequence of real numbers. Then lim sup xn is the largest limit of a subsequence of (xn ). Of course, a similar statement is true of the limit inferior.

Exercises 1. Find the limit superior and limit inferior of the following sequences. (a) (1/n) (b) (cos(nπ/2)) = (0, −1, 0, 1, 0, −1, 0, 1, . . .) (c) (2 + ((−1)n /n)) (d) ((n + 1)/n) 4 − 2−n , if n is even (e) 3, if n is odd 2. The rationals in (0, 1) are countable, say Q = (q1 , q2 , q3 , . . . ). Find lim inf Q and lim sup Q. 3. Fill in the details in the last two results.