Question 3
Consider pairs of 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
$$ A_{n}=a_{1}+\cdots+a_{n},\quad B_{n}=b_{1}+\cdots+b_{n};\qquad n=1,2,\ldots. $$
For any pair define $ c_i = \min\{a_i, b_i\} $ and $ C_n = c_1 + \cdots + c_n $, $ n = 1, 2, \ldots $.
(1) Does there exist a pair $(a_{i})_{i\geq1}$, $(b_{i})_{i\geq1}$ such that the sequences $(A_{n})_{n\geq1}$ and $(B_{n})_{n\geq1}$ are unbounded while the sequence $(C_{n})_{n\geq1}$ 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.
$$ a_{1}\geq a_{2}\geq a_{3}\geq\cdots,\qquad b_{1}\geq b_{2}\geq b_{3}\geq\cdots $$
and the sums
$$ A_{n}=a_{1}+\cdots+a_{n},\quad B_{n}=b_{1}+\cdots+b_{n};\qquad n=1,2,\ldots. $$
For any pair define $ c_i = \min\{a_i, b_i\} $ and $ C_n = c_1 + \cdots + c_n $, $ n = 1, 2, \ldots $.
(1) Does there exist a pair $(a_{i})_{i\geq1}$, $(b_{i})_{i\geq1}$ such that the sequences $(A_{n})_{n\geq1}$ and $(B_{n})_{n\geq1}$ are unbounded while the sequence $(C_{n})_{n\geq1}$ 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.