By Kenneth Hoffman

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But a and b do not belong to the set, and so it is not closed. The closure of (a, b), on the other hand, is [a, bJ. 32 1. Sets 12. The closed interval [a, b] is a elosed set. 13. The unit cirele S = {z E C: Izl = 1} is a elosed set in C; every member of S is a point of accumulation of S. Sequences of numbers. The concept of a sequence is central to analysis. In due course we deal with sequences in fairly arbitrary spaces; to set the stage for these later developments, and in order to be able to deal he re with some furt her ideas which are pertinent to real or complex numbers, we introduce sequences in this elementary setting.

The Bolzano-Weierstrass theorem. The quest ion naturally arises as to whether it is possible to eharaeterize those subsets for which all sequences contain a point of aecumulation in the subset. The answer, in the case of subsets of lR, is surprisingly straight forward: all that is required is for the set to be closed and bounded. This result is the substance of the following theorem. THEOREM 1 (THE BOLZANO-WEIERSTRASS THEOREM). Let [a,b] be a closed and bounded interval on the real line, and {x n } a sequence in [a, b].

So we have instead to be satisfied with an estimate of the error. To complete the pieture, one would hope to obtain from such an estimate not only an idea about the size of the error, but also some information about whether the approximatc procedure gives a family of solutions that converge to the exact solution and, if so, what thc rate 01 convergence iso 18 Introduction u(x) x FIGURE 4. The norm of a continuous function The finite element method is ideally suited to such a convergence analysis.