By Terence Tao
This is an element of a two-volume e-book on actual research and is meant for senior undergraduate scholars of arithmetic who've already been uncovered to calculus. The emphasis is on rigour and foundations of study. starting with the development of the quantity structures and set thought, the ebook discusses the fundamentals of research (limits, sequence, continuity, differentiation, Riemann integration), via to energy sequence, a number of variable calculus and Fourier research, after which ultimately the Lebesgue quintessential. those are virtually solely set within the concrete atmosphere of the genuine line and Euclidean areas, even if there's a few fabric on summary metric and topological areas. The publication additionally has appendices on mathematical good judgment and the decimal process. the total textual content (omitting a few much less critical issues) will be taught in quarters of 25–30 lectures each one. The direction fabric is deeply intertwined with the routines, because it is meant that the scholar actively examine the fabric (and perform considering and writing conscientiously) through proving a number of of the major ends up in the theory.
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Then the following are equivalent: • L is a limit point of (x(n) )∞ n=m . • There exists a subsequence (x(nj ) )∞ j=1 of the original sequence (n) ∞ (x )n=m which converges to L. Proof. 2. 8). 4. 6 (Cauchy sequences). Let (x(n) )∞ n=m be a sequence of points in a metric space (X, d). We say that this sequence is a Cauchy sequence iﬀ for every ε > 0, there exists an N ≥ m such that d(x(j) , x(k) ) < ε for all j, k ≥ N . 7 (Convergent sequences are Cauchy sequences). Let (x(n) )∞ n=m be a sequence in (X, d) which converges to some limit x0 .
Let n, m ≥ 0 be integers. Suppose that for every 0 ≤ i ≤ n and 0 ≤ j ≤ m we have a real number cij . Form the function P : R2 → R deﬁned by n m cij xi y j . ) Show that P is continuous. ) Conclude that if f : X → R and g : X → R are continuous functions, then the function P (f, g) : X → R deﬁned by P (f, g)(x) := P (f (x), g(x)) is also continuous. 2. 6. Let Rm and Rn be Euclidean spaces. If f : X → Rm and g : X → Rn are continuous functions, show that f ⊕ g : X → Rm+n is also continuous, where we have identiﬁed Rm × Rn with Rm+n in the obvious manner.
Proof. 5. 1. Let (X, ddisc ) be a metric space with the discrete metric. Let E be a subset of X which contains at least two elements. Show that E is disconnected. 2. Let f : X → Y be a function from a connected metric space (X, d) to a metric space (Y, ddisc ) with the discrete metric. Show that f is continuous if and only if it is constant. 3. 5. 4. 6. 5. 7. 6. Let (X, d) be a metric space, and let (Eα )α∈I be a collection of connected sets in X. Suppose also that α∈I Eα is non-empty. Show that α∈I Eα is connected.