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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 ?

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