By Hugo D. Junghenn

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The squeeze principle then implies that an , bn → x. 8 Definition. (Infinite limits) A sequence {an } in R is said to diverge to +∞, written an → +∞ or lim an = lim an = +∞, n n→+∞ if for each real number M there exists an index N such that an ≥ M for all n ≥ N . Divergence to −∞ is defined analogously. 9 Example. If r > 1, then rn /n → +∞. 6: Given M > 0 there exists N ∈ N such that n/rn < 1/M , hence rn /n > M , for all n ≥ N. 10 Example. If r > 0, then an := rn n! → +∞. Indeed, since an = rn → +∞, an−1 there exists N ∈ N such that an > 2an−1 , for all n > N .

From the completeness property one may deduce (Exercise 1) the symmetrical property Every nonempty subset of R that is bounded below has a greatest lower bound. The real number system may now be described as a complete ordered field. It may be shown that (up to isomorphism) there is exactly one such structure. The following important consequence of completeness is useful in determining the infimum or supremum of certain sets. It asserts that positive integer multiples of a positive real number may be made arbitrarily large.

Yn ). The Euclidean inner product x · y of x and y and the Euclidean norm x 2 of x are defined by n n 1/2 √ x·y = xj yj and x 2= x2j = x · x. j=1 j=1 The set R with its vector space structure and the Euclidean inner product is called n-dimensional Euclidean space. ♦ n The structure of Euclidean space allows one to define lines, planes, length, perpendicularity, angle between vectors, etc. These ideas will be useful in later chapters. 3 Theorem. The inner product in Rn has the following properties: (a) x · x = x 22 .