MathDB

Consider pairs of the sequences of positive real numbers

a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots

and the sums A_n = a_1 + \cdots + a_n, B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots. For any pair define

c_n = \min\{a_i,b_i\}

and

C_n = c_1 + \cdots + c_n

n=1,2,\ldots

.
(1) Does there exist a pair

(a_i)_{i\geq 1}

(b_i)_{i\geq 1}

such that the sequences

(A_n)_{n\geq 1}

and

(B_n)_{n\geq 1}

are unbounded while the sequence

(C_n)_{n\geq 1}

is bounded?
(2) Does the answer to question (1) change by assuming additionally that

b_i = 1/i

i=1,2,\ldots

?
Justify your answer.

Problem Statement