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Exercise 1: Positive number A

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exercise 1:
Let an be a sequence which converges to a positive number A. We showed in class that there
is an N in N such that for all n > N in N, |an| >
A
2
. From there, show that 1
an
converges
to
1
A
.
exercise 2:
Optional 2.6.G from Davidson – Donsig.
exercise 3:
Prove or disprove:
Let an be a sequence of real numbers. If limn→∞
(an+1 − an) = 0, then an is convergent.
exercise 4:
Let q be a fixed positive number. Show that the sequence an =
q
n
n!
is eventually decreasing.
exercise 5:
2.6.B from Davidson – Donsig. Hint: set f(x) = √
5 + 2x. Solve f(x) = x and the inequality
x ≤ f(x). Prove by induction that 0 ≤ an ≤ an+1 ≤ 1 + √
6.
exercise 6:
2.7.A
exercise 7:
2.7.G. Hint: any integer p can written as 3n − 1, 3n, or 3n + 1.

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