By Terence Tao

This is often half one among a two-volume creation to actual research and is meant for honours undergraduates, who've already been uncovered to calculus. The emphasis is on rigour and on foundations. the fabric starts off on the very starting - the development of quantity platforms and set concept, then is going directly to the fundamentals of research (limits, sequence, continuity, differentiation, Riemann integration), via to strength sequence, numerous variable calculus and Fourier research, and at last to the Lebesgue quintessential. those are virtually totally set within the concrete atmosphere of the true line and Euclidean areas, even though there's a few fabric on summary metric and topological areas. There are appendices on mathematical good judgment and the decimal process. the complete textual content (omitting a few much less vital issues) will be taught in quarters of twenty-five to thirty lectures every one. The direction fabric is deeply intertwined with the routines, because it is meant that the coed actively research the fabric (and perform considering and writing conscientiously) by means of proving a number of of the foremost leads to the speculation. the second one variation has been widely revised and up-to-date.

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In what follows we let (X, d) be a metric space. (a) Given any Cauchy sequence (xn)~=l in X, we introduce the formal limit LIMn-+ooXn· We say that two formal limits LIMn-+ooXn and LIMn-+ooYn are equal if limn-+oo d(xn, Yn) is equal to zero. Show that this equality relation obeys the reflexive, symmetry, and transitive axioms. (b) Let X be the space of all formal limits of Cauchy sequences in X, with the above equality relation. Define a metric dx : X x X ---t R + by setting lim d(xn, Yn)· dx(LIMn-+ooXn, LIMn-+ooYn) := n-+oo Show that this function is well-defined (this means not only that the limit limn-+oo d(xn, Yn) exists, but also that the axiom of substitution is obeyed; cf.

The set B(y(l), ro/2) U B(yC 2 ), r 0 /2) cannot cover all of Y, since we would then obtain two sets V01 and V02 which covered Y, a contradiction again. So we can choose a point y( 3 ) which does not lie in B(yC 1 ), ro/2) U B(yC 2 ), ro/2), so in particular d(yC 3 ), yC 1l) ~ ro/2 and d(y( 3 ), yC 2 l) ~ ro/2. Continuing in this fashion we obtain a sequence (y(n))~=l in Y with the property that d(y(k), y0l) ~ ro/2 for all k ~ j. In particular the sequence (y(n))~=l is not a Cauchy sequence, and in fact no subsequence of (y(n))~=l can be a Cauchy sequence either.

6 (Compact sets are closed and bounded). Let (X, d) be a metric space, and let Y be a compact subset of X. Then Y is closed and bounded. 414 12. Metric spaces The other half of the Heine-Borel theorem is true in Euclidean spaces: Theorem 12. 5. 7 (Heine-Borel theorem). Let (Rn, d) be a Euclidean space with either the Euclidean metric, the taxicab metric, or the sup norm metric. Let E be a subset of R n. Then E is compact if and only if it is closed and bounded. Proof. 3. D However, the Heine-Borel theorem is not true for more general metrics.