1) (i) Prove that every sequence converges to every point in a space with the trivial topology. (ii) Prove that no sequence converges to any point in a space with the discrete topology.

Q: Theorem10.2 is All bounded monotone sequence converge.

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Q: Prove the sequence converges and find its limit.

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Q: Prove that if a sequence  {an} converges, it must be bounded. Is the converse true as well

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Q: Find an example of two unbounded sequences , such that their product gives a bounded sequence.

A: Consider the two unbounded sequences

Q: 그 n=1

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Q: Prove that each convergent sequence of reals is bounded.

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Q: A first countable space is a hausdroff space iff every convergent sequence has a unique limit

A: Let X=S,τ be first countable Hausdorff space. Let <xn> be a convergent sequence in X. To show:…

Q: nx Pointwise Show that { J 1+ 3n on [1,00) but not uniform. Hint: use Xn = 1+ 3n = f(x)

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Q: Every convergent sequence of real numbers is not bounded ylgn İhi

A: Every convergent sequence of real numbers is not bounded.

Q: 14. Prove that every Cauchy sequence is bounded (Theorem 1.4)

A: To prove: Every Cauchy sequence is bounded.

Q: B. Prove Bolzano-Weierstrass Theorem. a bounded sequence of real numbers has a convergent…

A: We have to prove

Q: 5. Prove that every bounded complex sequence has a convergent subsequence

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Q: n! 1 · 3 · 5 ··· (2n − 1). +∞ n=1

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Q: Prove that there is no sequence of real numbers whose set of accumulation points is exactly Q

A: Answer given below.

Q: 10.2 Theorem. All bounded monotone sequences converge. Proof Let (sn) be a bounded increasing…

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Q: 2. Prove that every bounded sequence in a metric space has a convergent subsequence

A: To prove that every bounded sequence in a metric space has a convergent subsequence. It is enough to…

Q: 2. a) Suppose (pn) has limit L, and let S be the set of values of the terms of (pn). Can we say that…

A: a) Suppose pn has a limit say L. That is, pn→L. For given ε>0, there exists n0∈ℕ such that…

Q: According to the Bolzano-Weierstrass theorem, every bounded sequence of real numbers has a…

A: Solution :- True Explanation :-

Q: Let (x₂) be an unbounded and increasing sequence of real numbers. Prove that x (x₂) diverges to +…

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Q: Show that a sequence (an) is bounded if and only if it is bounded above and below.

A: Sequence is bounded if and only if it is bounded below and bounded above.Explanation:

Q: Prove that every bounded below, decreasing sequence converges

A: Let {an} be a monotonically decreasing sequence which is bounded below.  Since the sequence is…