3. Let xn =
2n+1
3n+7 .
(a) Prove, directly using the definition, that limn→∞
xn =
2
3
.
(b) Prove, using the algebraic limit theorem, that limn→∞
xn =
2
3
8. (a) Let (xn) be bounded (not necessarily convergent) and assume that yn → 0 as
n → ∞. Show that xnyn → 0 as n → ∞. (Why can we not just use the Algebraic
limit theorem?)
(b) What about if yn → y as n → ∞ where y 6= 0?
10. For the following, provide an example or prove that no such request is possible. You
may appeal to results from lectures.
(a) Sequences (xn) and (yn) which both diverge, but whose sum (xn + yn) converges.
(b) Sequences (xn), which converges, and (yn), which diverges, but whose sum (xn+yn)
converges.
(c) A convergent sequence (xn), such that xn 6= 0 for all n ∈ N and (1/xn) diverges.
(d) An unbounded sequence (xn) and a convergent sequence (yn) with (xn−yn) bounded.
(e) two sequences (xn) and (yn), where (xnyn) and (xn) converge, but (yn) does not
converge.
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