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 . From there, show that converges to .
exercise 2:
Optional 2.6.G from Davidson - Donsig.
exercise 3:
Prove or disprove: Let an be a sequence of real numbers. If lim (an+1 − an) = 0, then an is convergent. n→∞
exercise 4:
Let q be a fixed positive number. Show that the sequence 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.
