Question 140
Let $m, n$ be positive integers. Consider a sequence of positive integers $a_{1}$, $a_{2}, \ldots, a_{n}$ satisfying $m=a_{1} \geq a_{2} \geq \cdots \geq a_{n} \geq 1$. For $1 \leq i \leq m$ we define
$$
b_{i}=\#\left\{j \in\{1,2, \ldots, n\}: a_{j} \geq i\right\}
$$
so $b_{i}$ is the number of terms $a_{j}$ of the sequence for which $a_{j} \geq i$. For $1 \leq j \leq n$ we define
$$
c_{j}=\#\left\{i \in\{1,2, \ldots, m\}: b_{i} \geq j\right\}
$$
so $c_{j}$ is the number of terms $b_{i}$ for which $b_{i} \geq j$.
Example: the a-sequence 5, 3, 3, 2, 1, 1 corresponds to the b-sequence 6, 4, 3, 1, 1 .
a. Prove that $a_{j}=c_{j}$ for $1 \leq j \leq n$.
b. Prove that for $1 \leq k \leq m$: $\sum_{i=1}^{k} b_{i}=k \cdot b_{k}+\sum_{j=b_{k}+1}^{n} a_{j}$.
$$
b_{i}=\#\left\{j \in\{1,2, \ldots, n\}: a_{j} \geq i\right\}
$$
so $b_{i}$ is the number of terms $a_{j}$ of the sequence for which $a_{j} \geq i$. For $1 \leq j \leq n$ we define
$$
c_{j}=\#\left\{i \in\{1,2, \ldots, m\}: b_{i} \geq j\right\}
$$
so $c_{j}$ is the number of terms $b_{i}$ for which $b_{i} \geq j$.
Example: the a-sequence 5, 3, 3, 2, 1, 1 corresponds to the b-sequence 6, 4, 3, 1, 1 .
a. Prove that $a_{j}=c_{j}$ for $1 \leq j \leq n$.
b. Prove that for $1 \leq k \leq m$: $\sum_{i=1}^{k} b_{i}=k \cdot b_{k}+\sum_{j=b_{k}+1}^{n} a_{j}$.