Advanced Analysis by Min Yan

By Min Yan

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X1 x2 x3 x4x5 b .. 16. Prove the inductively defined sequences converge and find the limits. 1. x1 = 1, xn+1 = 1 + 2 . xn 2. x1 = 1, xn+1 = x2n + 2 . 2. 17.

11. 1. A sequence with no upper bound must have a subsequence diverging to +∞. This means limn→∞ xn = +∞. 2. If there is no subsequence with finite limit and no subsequence diverging to −∞, then the whole sequence diverges to +∞. Supremum and Infimum in Ordered Set Recall that an order on a set is a relation x < y between pairs of elements satisfying the transitivity and the exclusivity. The concepts of upper bound, lower bound, supremum and infimum can be defined for subsets of an ordered set in a way similar to numbers.

A .. .. 4: for any , there is δ Similar to the limit of sequences, the predetermined smallness for |f (x)− l| is arbitrarily given, while the size δ for |x − a| is to be found after is given. Thus the choice of δ usually depends on and is often expressed as a function of .

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