Školjka

%V0 Consider two monotonically decreasing sequences $\left( a_k\right)$ and $\left( b_k\right)$, where $k \geq 1$, and $a_k$ and $b_k$ are positive real numbers for every k. Now, define the sequences $c_k = \min \left( a_k, b_k \right)$; $A_k = a_1 + a_2 + ... + a_k$; $B_k = b_1 + b_2 + ... + b_k$; $C_k = c_1 + c_2 + ... + c_k$ for all natural numbers k. (a) Do there exist two monotonically decreasing sequences $\left( a_k\right)$ and $\left( b_k\right)$ of positive real numbers such that the sequences $\left( A_k\right)$ and $\left( B_k\right)$ are not bounded, while the sequence $\left( C_k\right)$ is bounded? (b) Does the answer to problem (a) change if we stipulate that the sequence $\left( b_k\right)$ must be $\displaystyle b_k = \frac {1}{k}$ for all k ?