| Chapter 2, Section 1 |
Theorem 2.1.1 | Rn is a vector space |
| Theorem 2.1.2 | The Euclidean dot product is an inner product |
| Theorem 2.1.3 | The Euclidean norm in Rn is a norm |
| Theorem 2.1.4 | The l1 norm in Rn is a norm |
| Theorem 2.1.5 | The sup norm in Rn is a norm |
| Theorem 2.1.6 | The Euclidean distance in Rn is a metric |
| Theorem 2.1.7 | (8.5) The Cauchy-Schwartz inequality |
| Theorem 2.1.8 | Bounds for the Euclidean norm in terms of the sup norm |
| Theorem 2.1.9 | Bounds for the l1 norm in terms of the Euclidean norm |
| Theorem 2.1.10 | (8.6iii) Second form of triangle inequality for Euclidean norm |
| Chapter 2, Section 2 |
Theorem 2.2.1 | (8.21) An open ball in Rn is an open set |
| Theorem 2.2.2 | (8.22) Singleton sets in Rn are closed |
| Theorem 2.2.3 | (8.23) The empty set is both open and closed |
| Theorem 2.2.4 | (8.23) Rn is both open and closed |
| Theorem 2.2.5 | The open sets defined previously constitute a topology for Rn |
| Theorem 2.2.6 | (8.32i)For any subset E of Rn, E contains its interior, and its closure contains E |
| Theorem 2.2.7 | (8.32ii) If V is an open subset of E, then V is a subset of the interior of E |
| Theorem 2.2.8 | (8.32iii) If F is closed and F contains E, then F contains the closure of E |
| Theorem 2.2.9 | (8.36) The boundary of E is the closure of E setminus the interior of E |
| Theorem 2.2.10 | (8.37ia)The interior of the union of A and B contains the union of their interiors |
| Theorem 2.2.11 | (8.37iib) The intersection of the closures of A and B contains the closure of their intersection |
| Theorem 2.2.12 | (8.37ib)The intersection of the interiors of A and B is the interior of their intersection |
| Theorem 2.2.13 | (8.37iia)The union of the closures of A and B is the closure of their union |
| Theorem 2.2.15 | (8.37iiia) The union of the boundaries of A and B contains the boundary of their union |
| Theorem 2.2.16 | (8.37iiib) The intersection of the boundaries of A and B contains the boundary of their intersection |
Chapter 3, Section 1 |
Theorem 3.1.1 | (9.2) A sequence in Rn converges if and only if its component sequences converge in R |
| Theorem 3.1.2 | (9.3) Every element of Rn is a limit of a sequence in Qn |
| Theorem 3.1.3 | Rn is separable |
| Theorem 3.1.4 | (9.4i) A sequence in Rn can have at most one limit |
| Theorem 3.1.5 | (9.4ii) If a sequence in Rn converges, all of its subsequences converge, and to the same limit |
| Theorem 3.1.6 | (9.4iii) Every convergent sequence in Rn is bounded. The converse is false. |
| Theorem 3.1.7 | (9.4iv) Every convergent sequence in Rn is Cauchy |
| Theorem 3.1.8 | (9.4va) The limit of a sum of two sequences is the sum of the limits, if they exist |
| Theorem 3.1.9 | (9.4vb) The limit of a constant times a sequence is the constant times the limit of the sequence, if it exists |
| Theorem 3.1.10 | (9.4vc)The limit of the product of two sequences is the product of their limits, if they exist |
| Theorem 3.1.11 | The limit of the norms of the terms of a convergent sequence in Rn equals the norm of the limit |
| Theorem 3.1.12 | (9.5 Bolzano-Weierstrass) Every bounded sequence in Rn has a convergent sequence |
| Theorem 3.1.13 | (9.6) A sequence in Rn is convergent if and only if it is Cauchy |
| Theorem 3.1.14 | (9.7) A sequence in Rn converges to a if and only if for every open set V containing a, there is an N such that xk is in V for all k>N |
| Theorem 3.1.15 | (9.8) A subset of Rn is closed if and only if it contains its limit points |
| Chapter 3, Section 2 |
Theorem 3.2.1 | (9.9) The Borel covering lemma |
| Theorem 3.2.2 | (9.11) The Heine-Borel theorem |
| Chapter 3, Section 3 |
Theorem 3.3.1 | (9.15i) If f(x)=g(x) except at x=a and lim f(x) exists, lim g(x)=lim f(x) |
| Theorem 3.3.2 | (9.15ii) Sequential characterization of limits |
| Theorem 3.3.3 | (9.15iiia) lim(f+g)(x)=lim f(x)+lim g(x) |
| Theorem 3.3.4 | (9.15iiib) lim(af)(x)=a lim f(x) |
| Theorem 3.3.5 | (9.15iiic) lim(fg)(x)= lim f(x) lim g(x) |
| Theorem 3.3.6 | (9.15iiid) The norm of lim f(x) equals the limit of the norm f(x) |
| Theorem 3.3.7 | (9.15iv) Squeeze theorem for functions |
| Theorem 3.3.8 | (9.15v) Limit of a composition |
| Theorem 3.3.9 | (9.16) The limit of vector is the vector of component limits |
| Chapter 4, Section 1 |
Theorem 4.1.1 | (Example 10.2) Rn with the usual metric is a metric space |
| Theorem 4.1.2 | (Example 10.3) R with the discrete metric is a metric space |
| Theorem 4.1.3 | (Example 10.4) If E is a subset of X, (E,rho) is a metric space |
| Theorem 4.1.4 | (Example 10.5) (Q,rho) is a metric space with rho(x,y)=|x-y| |
| Theorem 4.1.5 | (Example 10.6) The space of continuous functions on [a,b] is a metric space with rho(f,g)=||f-g|| and ||f||=sup |f(x)| |
| Theorem 4.1.6 | (Example 10.9a) Every open ball in (X,rho) is open |
| Theorem 4.1.7 | (Example 10.9b) Every closed ball in (X,rho) is closed |
| Theorem 4.1.8 | (Example 10.10) Singleton sets in (X,rho) are closed |
| Theorem 4.1.9 | (Remark 10.11) In a metric space (X,rho), X and the empty set are both open and closed |
| Theorem 4.1.10 | (Example 10.12) Every subset of the discrete metric space (R,sigma) is open |
| Theorem 4.1.11 | (Theorem 10.14i) A sequence in a metric space (X,rho) can have at most one limit |
| Theorem 4.1.12 | (Theorem 10.14ii) If xn converges to a, so does every subsequence of xn |
| Theorem 4.1.13 | (Theorem 10.14iii) Every convergent sequence in a metric space is bounded |
| Theorem 4.1.14 | (Theorem 10.14iv) Every convergent sequence in a metric space is Cauchy |
| Theorem 4.1.15 | (Remark 10.15) xn converges to a iff for every open set containing a, xn belongs to V when n >=N |
| Theorem 4.1.16 | (Theorem 10.16) A subset E of X is closed if and only if the limit of every convergent sequence in E belongs to E |
| Theorem 4.1.17 | (Theorem 10.17) The discrete space (R,sigma) contains bounded sequences that have no convergent subsequence |
| Theorem 4.1.18 | (Remark 10.18) The metric space (Q,rho) contains Cauchy sequences that do not converge |
| Theorem 4.1.19 | (Theorem 10.21) A subset E of the complete metric space (X,rho) is itself a complete metric space if and only if it is closed. |
| Chapter 4, Section 2 |
Theorem 4.2.1 | (Theorem 10.26i) If f and g are identical except at x=a and lim f(x) exists as approaches a, lim g(x) is the same |
| Chapter 4, Section 3 |
Theorem 4.3.1 | (Remark 10.43) The empty set and all finite subsets of a metric space are compact |
| Theorem 4.3.2 | (Remark 10.44) All compact sets are closed |
| Theorem 4.3.3 | (Remark 10.45) A closed subset of a compact set is closed |
| Theorem 4.3.4 | (Theorem 10.46) A compact subset of a metric space is closed and bounded |
| Theorem 4.3.5 | (Remark 10.47) The converse of the previous statement is false |
| Theorem 4.3.6 | (10.49 Lindelof) Every open cover of a separable metric space has a countable subcover |
| Theorem 4.3.7 | (10.50 Heine-Borel) A subset H of a separable metric space with the Bolzano-Weierstrass property is compact iff it is closed and bounded |
| Chapter 4, Section 4 |
Theorem 4.4.1 | (10.52) A continuous function on a compact subset H is uniformly continuous on H |
| Theorem 4.4.2 | (10.58) f is continuous iff the inverse image of an open set is an open set |
| Theorem 4.4.3 | (10.61) The image of a compact set under a continuous function is compact |
| Theorem 4.4.4 | (10.63) Extreme Value Theorem |
| Theorem 4.4.5 | (10.64) If H is compact and f continuous, 1-1, f-inverse is continuous on f(H) |
| Theorem 4.4.6 | (10.69 Stone-Weierstrass) On a compact metric space an algebra that separates points and contains the constant functions is uniformly dense |