Point Set Theory and Sequences
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1.) Let 'S" be the set of points on the curve y = Sin (1/x) (X #0) in the XY -Plane.
(a) Find a sequence of point Pn on the X - axis and in "S" such that Pn converges to (0, 0).
(b) Find a sequence of pints Pn on the line Y = 1 and in "S" such that Pn coverges to (0, 1).
(c) What points must be adjoined to "S" in order to get a closed set?
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1.) Let a sequence be defined on the X - axis as follows: X1 = 1, X2 = 2, X3 = ½(X1 + X2) and in general Xn+1 = ½ (Xn-1 + Xn), n = 2, 3, ...... Show that ∣Xm - Xn ∣≤ ½ N-1 If N ≤ m and N ≤ n, so that Cauchy's condition is fulfilled. SUGGESTION: Note that each term is midway between the two preceding terms.
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Solution Summary
Point Set Theory and Sequences are investigated. The solution is detailed and well presented.
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