MATH 111

HOMEWORK 5

1) Let (a_n)_N and (b_n)_N be two convergent sequences with limits A and B, respectively. Prove that

a) The sequence (a_n + b_n)_N converges to A+B,

b) The sequence (c.a_n)_N converges to c.A , where c is a constant,

c) The sequence (a_nb_n)_N converges to A.B.

( Hint: Use the identity 2a_nb_n = (a_n+b_n)²- a_n² - b_n² )

2) The following sequences are convergent. Find their limits, justify your answer.

a) a_n = 1/n

b) a_n = n/(n+1)

c) a_n = 1/n!

d) a_n = 2n/(n³+1)

e) a_n= (-1)ⁿ⁺¹ /n.

3) A sequence (a_n)_N is bounded if there is some M s.t. ½a_n½< M for all natural number n.

Show that every Cauchy sequence is bounded.

4) Show that every convergent sequence is Cauchy.

5) Let (a_n)_N and (b_n)_N be two Cauchy sequences. Prove that the sequence (a_n + b_n)_N and the sequence (a_nb_n)_N are Cauchy sequences.