prove that an is bounded above.

Q: Suppose A ⊂ R is both closed and bounded, prove that A has the Heine-Borel Property

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Q: Consider the set T = {..., −34, −17, 0, 17, 34, ... } with the usual definition of less than (a0).…

A: Completeness property of real numbers:  Every non-empty set of real numbers that has an upper bound…

Q: Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded

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Q: Let f: R -> R and E ⊆ R. Prove f is continous iff f-1(E) is closed whenever E ⊆ R is closed

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Q: Let S be a nonempty subset of R that is bounded below. Prove that inf S= -sup{-s:s∊S}

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Q: ► Exercise 4.4. Prove that a topological space X is disconnected if and only there exists a…

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Q: Prove that the set {x e R : 10/x - x > 0} is bounded. Prove that the set {x E R : x² – 25x > 0} is…

A: To prove that the set x∈R:10x−x>0 is bounded. To prove that the set x∈R:x2−25x>0 is unbounded.

Q: Prove that if f : R → R is continuous, then the image of a bounded set over f is bounded.

A: Given f:R->R is continuous function  Let A is bounded subset of R  So the limit point set A' is…

Q: sketch the set {z ∈ C : Re(z) = Im(z + 2 + i)}

A: We have to sketch the set {z∈ℂ:Re(z)=Im(z+2+i)} Let z=x+iy where x,y∈ℝ

Q: Suppose A is a closed, non-empty subset of a metric space (X, d) and p a point of X which is not in…

A: Given:  A is a closed and non-empty set of a metric space (X,d) and p∈X , p∉A. To Prove: inf{d(p,a)…

Q: Let A = Z. Let R be defined on A x A where a Ry if 4 (x- y). Prove that R is transitive.

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Q: Let us assume the facts for all a, 6 E R, i.e. as < bi whenever a < b. Let SCR be bounded above and…

A: Given that S~=x∈ℝ|x3∈S And S⊂ℝ is bounded above. We say that S⊂ℝ is bounded above if  and if…

Q: t & · &1 and & · &2 be two equivalent norms on a vector space X. Prove that a sequence % xn & in X…

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Q: We say that a set A is denumerable if there exists a bijective function f : N→ A. Suppose that A and…

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Q: Prove that the boundary points of an open interval (a, b) CR are the points a and b.

A: A point a∈ℝ is said to be boundary point of a set A⊂ℝ, if Ba,r∩A≠∅ and Ba,r∩Ac≠∅ for every…

Q: ": Prove that ( R", ||||) is complete metric space.

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