Use the Heine-Borel Theorem to prove the Bolzano-Weierstrass Theorem.
Use the Heine-Borel Theorem to prove the Bolzano-Weierstrass
Theorem.
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Use the Heine-Borel Theorem to prove the Bolzano-Weierstrass
Theorem.
Use the Heine-Borel Theorem to prove the Bolzano-Weierstrass The- orem.
Use the Heine-Borel Theorem to prove the Bolzano-Weierstrass The- orem.
Problem 5. Show that the assertion of the Heine - Borel theorem is equivalent to the...
Problem 5. Show that the assertion of the Heine - Borel theorem is equivalent to the Com- pleteness Ariom for the real numbers.
2. (8 pts) In contrast to the Heine-Borel Theorem in R", show that the closed unit...
2. (8 pts) In contrast to the Heine-Borel Theorem in R", show that the closed unit ball in f = {(21,..., In, ...): -10 < } is not compact. (Hint: find a sequence on the unit sphere containing no convergent subsequence)
(a) By the Heine-Borel Theorem, show that R2 is not compact and the sphere S2 ={(x,y,z)∈R3 :x2 +y2 +z2 =1} is compact in...
(a) By the Heine-Borel Theorem, show that R2 is not compact and
the
sphere
S2 ={(x,y,z)∈R3 :x2 +y2 +z2 =1}
is compact in R3.
(b) Show that R2 and S2 is not homeomorphic. (i.e. no continuous
bi-
jective function f between R2 and S2 such that the inverse function
f−1 is continuous).
Question 1. (2 marks) (a) By the Heine-Borel Theorem, show that R2 is not compact and the sphere is compact in R3. (b) Show that R2 and S2...
THE BOLZANO-WEIERSTRASS THEOREM 2.4.1 Determine which of the following are Cauchy sequences. (a) an = (-1)"...
THE BOLZANO-WEIERSTRASS THEOREM
2.4.1 Determine which of the following are Cauchy sequences. (a) an = (-1)" (b) An = (-1)"/n. (c) an=n/(n+1). (d) an = (cosn)/n.
A. Assume the Weierstrass Theorem is true for C0, 1, and then prove it is true...
A. Assume the Weierstrass Theorem is true for C0, 1, and then prove it is true for C[a, b, for an arbitrary interval la, b HINT: For f E Cla, b), consider g(t)f(a+(b-a)t) in C0, 1
A. Assume the Weierstrass Theorem is true for C0, 1, and then prove it is true for C[a, b, for an arbitrary interval la, b HINT: For f E Cla, b), consider g(t)f(a+(b-a)t) in C0, 1
(a) Suppose that lim x→c f(x) = L > 0. Prove that there exists a δ...
(a) Suppose that lim x→c f(x) = L > 0. Prove that there
exists a
δ > 0 such that if 0 < |x − c| < δ, then f(x) >
0.
(b) Use Part (a) and the Heine-Borel Theorem to prove that if
is
continuous on [a, b] and f(x) > 0 for all x ∈ [a, b], then
there
exists an " > 0 such that f(x) ≥ " for all x ∈ [a, b].
= (a) Suppose...
prove e EOLU Exercise 4.1.1. Prove Theorem 4.1.6. (Hints: for (a) and (b), use the root...
prove e
EOLU Exercise 4.1.1. Prove Theorem 4.1.6. (Hints: for (a) and (b), use the root test (Theorem 7.5.1). For (c), use the Weierstrass M-test (Theorem 3.5.7). For (d). use Theorem 3.7.1. For (e), use Corollary 3.6.2. The signale UI tre rauUS UI CUNvergence is the IUIUWII. Theorem 4.1.6. Let - Cn(x-a)" be a formal power series, and let R be its radius of convergence. (e) (Integration of power series) For any closed interval [y, z] con- tained in (a...
4. (a) Suppose that limz-c f(x) = L > 0. Prove that there exists a 8...
4. (a) Suppose that limz-c f(x) = L > 0. Prove that there exists a 8 >0 such that if 0 < 12 – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on (a, b) and f(x) > 0 for all x € (a, b), then there exists an e > 0 such that f(x) > € for all x € [a, b].
= (a) Suppose that limx+c f(x) L > 0. Prove that there exists a 8 >0...
= (a) Suppose that limx+c f(x) L > 0. Prove that there exists a 8 >0 such that if 0 < \x – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x € [a,b], then there exists an e > 0 such that f(x) > e for all x E [a, b].
Use the Heine-Borel Theorem to prove the Bolzano-Weierstrass The- orem.
Problem 5. Show that the assertion of the Heine - Borel theorem is equivalent to the Com- pleteness Ariom for the real numbers.
2. (8 pts) In contrast to the Heine-Borel Theorem in R", show that the closed unit ball in f = {(21,..., In, ...): -10 < } is not compact. (Hint: find a sequence on the unit sphere containing no convergent subsequence)
(a) By the Heine-Borel Theorem, show that R2 is not compact and
the
sphere
S2 ={(x,y,z)∈R3 :x2 +y2 +z2 =1}
is compact in R3.
(b) Show that R2 and S2 is not homeomorphic. (i.e. no continuous
bi-
jective function f between R2 and S2 such that the inverse function
f−1 is continuous).
Question 1. (2 marks) (a) By the Heine-Borel Theorem, show that R2 is not compact and the sphere is compact in R3. (b) Show that R2 and S2...
THE BOLZANO-WEIERSTRASS THEOREM
2.4.1 Determine which of the following are Cauchy sequences. (a) an = (-1)" (b) An = (-1)"/n. (c) an=n/(n+1). (d) an = (cosn)/n.
A. Assume the Weierstrass Theorem is true for C0, 1, and then prove it is true for C[a, b, for an arbitrary interval la, b HINT: For f E Cla, b), consider g(t)f(a+(b-a)t) in C0, 1
A. Assume the Weierstrass Theorem is true for C0, 1, and then prove it is true for C[a, b, for an arbitrary interval la, b HINT: For f E Cla, b), consider g(t)f(a+(b-a)t) in C0, 1
(a) Suppose that lim x→c f(x) = L > 0. Prove that there
exists a
δ > 0 such that if 0 < |x − c| < δ, then f(x) >
0.
(b) Use Part (a) and the Heine-Borel Theorem to prove that if
is
continuous on [a, b] and f(x) > 0 for all x ∈ [a, b], then
there
exists an " > 0 such that f(x) ≥ " for all x ∈ [a, b].
= (a) Suppose...
prove e
EOLU Exercise 4.1.1. Prove Theorem 4.1.6. (Hints: for (a) and (b), use the root test (Theorem 7.5.1). For (c), use the Weierstrass M-test (Theorem 3.5.7). For (d). use Theorem 3.7.1. For (e), use Corollary 3.6.2. The signale UI tre rauUS UI CUNvergence is the IUIUWII. Theorem 4.1.6. Let - Cn(x-a)" be a formal power series, and let R be its radius of convergence. (e) (Integration of power series) For any closed interval [y, z] con- tained in (a...
4. (a) Suppose that limz-c f(x) = L > 0. Prove that there exists a 8 >0 such that if 0 < 12 – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on (a, b) and f(x) > 0 for all x € (a, b), then there exists an e > 0 such that f(x) > € for all x € [a, b].
= (a) Suppose that limx+c f(x) L > 0. Prove that there exists a 8 >0 such that if 0 < \x – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x € [a,b], then there exists an e > 0 such that f(x) > e for all x E [a, b].
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