4. (10%) Consider a fixed positive integer k and a sequence of positive real numbers $a_1,\cdots,a_k,b_1,\cdots,b_k$, where $a_i>0, b_i>1$ for $1 \le i \le k$, such that $\sum_{i=1}^{k} a_i/b_i^r = 1$.

(A) [5%] Prove such r exists.

(B) [5%] Design an efficient procedure to find the real number r with error within $10^{-6}$. (Note: If you don't answer part (A) correctly, then you will not earn any point from this problem!)