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17 THEOREM [COMPARISON THEOREMj. Suppose that {xn} and {Yn} are convergent sequences. If there is an No E N such that (1) for n 2: No, then lim xn:S: lim Yn. n-+oo n-+oo In particular, if Xn E [a, bj converges to some point c, then c must belong to [a, bj. PROOF. , that (1) holds but x := liffin->oo Xn is greater than Y := limn->oo Yn. Set € = (x - y)/2. Choose Nl > No such that IX n - xl < € and IYn - YI < € for n 2: N 1 . Then f~r ~uch an n, Xn > X - € = X-Y X-Y X - -2- = Y + -2- = Y + € > Yn, which contradicts (1).

Oo Xn +(0) if and only if for each MER there is an N E N such that n ~ N implies Xn > M. (ii) {xn} is said to diverge to -00 (notation: Xn ----t -00 as n ----t 00 or limn ..... oo Xn -(0) if and only if for each MER there is an N E N such that n ~ N implies Xn < M. , eventually Xn exceeds every number M (no matter how large and positive M is). Similarly, Xn ----t -00 if and only if Xn eventually is less than every number M (no matter how large and negative Mis). It is easy to see that the Squeeze Theorem can be extended to infinite limits (see Exercise 6).

14 Lemma. lin, kEN and 1:::; k :::; n, then (n + k 1) = ( k-1 n ) + (n) k . (n) k = n! k! k! = PROOF. k! (n + k 1) . I Binomial coefficients can be used to expand the nth power of a sum of two terms. 15 THEOREM [BINOMIAL FORMULA]. lia,b E Rand n E N, then PROOF. The proof is by induction on n. The formula is obvious for n = 1. Suppose that the formula is true for some n E N. , the formula is true for n + 1. We conclude by induction that the formula holds for all n EN. 2 17 Well- Ordering Principle EXERCISES 00.

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Algorithms, graphs, and computers by R Cooke, K L. Lockett, J A Bellman

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