If $ \sum_{n=1}^{\infty} a_n $ is absolutely convergent, then every rearrangement of $ \sum_{n=1}^{\infty} a_n $ is also absolutely convergent?

Question

This question was asked before but my proof is different from that proof. So will you please check my proof? Let $ \sum_{n=1}^{\infty} a_n $ be an absolutely convergent series then every partial sum of $ \sum_{n=1}^{\infty} |a_n| $ is bounded by some $M>0$. Let $ \sum_{k=1}^{\infty} b_k $ be a rearrangement of $ \sum_{n=1}^{\infty} a_n $. Now consider the kth partial sum of $ \sum_{k=1}^{\infty} |b_k| $ that is $ s_k = |b_1|+|b_2|+\cdots+|b_k| $ since both the sequences $ (a_n) $ and $ ( b_n) $ have same elements but in different arrangement so we can write $ s_k=|a_{p_1}|+|a_{p_2}|+\cdots+|a_{p_k}| $. Let the greatest index be $p_j$, where $ 1\le j \le k $, then $ s_k \le |a_1|+|a_2|+\cdots+|a_{p_j}| \lt M$. Hence the proof.