Let {an}n≥1 be a convergent sequence of real numbers. 1) Show that if for all but finitely many an we have an ≥ a, then limn→∞ an ≥ a. 2) Show that if for all but finitely many an we have an ≤ b, then limn→∞ an ≤ b. 3) Conclude that if all but finitely many an belong to the interval [a, b], then limn→∞ an ∈ [a, b].

Q: Exercise 10. Show that if AC B, then ACB.

A:

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A: We can solve this question using definition of converges diverges and oscillating of sequences

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Q: 2. Prove that the sequence {n} defined by xn := (-1)n 3n+2 is divergent. Use an argument similar to…

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Q: (ii) If (sn) is an unbounded decreasing sequence, then lim så -∞

A: Given that {sn} is an unbounded decreasing sequence. To show :- limn→∞sn=-∞

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.