Why does the equation in the description hold (it was too long for the title) Let $X$ be a discrete random variable such that $\mathbb{P}(\{X \in \mathbb{N}\}) =1$. Besides, $(a_n)_{n \in \mathbb{N}} \in [0, \infty)^{\mathbb{N}}$. 
Why is $$\sum_{j=1}^{n} (a_1 + \ldots + a_j)\mathbb{P}(\{X=j\}) + \mathbb{P}(\{X \geq n + 1\})\sum_{j=1}^{n}a_j = \sum_{j=1}^{n} a_j \mathbb{P}(\{X \geq j\})?$$
for all $n \in \mathbb{N}$.
I have no idea how to approach this problem. I've tried rewriting the left side of the equation to $$
\sum_{j=1}^{n} \sum_{i=1}^{j} a_i \mathbb{P}(\{X=j\}) + \mathbb{P}(\{X \geq n + 1\})\sum_{j=1}^{n}a_j = \sum_{j=1}^{n} a_j \mathbb{P}(\{X \geq j\})?
$$
But it doesn't seem to help me at all. I also found out that $$
\mathbb{P}(\{X \geq n + 1\}) = \sum_{k = n+1}^{\infty} \mathbb{P}(X = k)
$$
but I still can't find a solution.
 A: We can prove this by induction in $n$. For $n = 1$, the equation becomes
$$a_1 P(X = 1) + P(X \geq 2)a_1 = a_1 P(X \geq 1).$$
So, assume that the equation holds for a given $n$. Then
$$\begin{align}
&\sum_{j=1}^{n+1} (a_1 + \dots + a_j)P(X = j) + P(X \geq n+2) \sum_{j=1}^{n+1} a_j
\\=&\sum_{j=1}^{n} (a_1 + \dots + a_j)P(X = j) + (a_1 + \dots + a_{n+1})P(X = n+1) + P(X \geq n+2) \sum_{j=1}^{n} a_j
\\ &+ P(X \geq n+2) a_{n+1}
\\=&\sum_{j=1}^{n} (a_1 + \dots + a_j)P(X = j) + (a_1 + \dots + a_{n+1})P(X = n+1) + P(X \geq n+1) \sum_{j=1}^{n} a_j
\\&- P(X = n+1) \sum_{j=1}^n a_j + P(X \geq n+2) a_{n+1} 
\\=&\sum_{j=1}^n a_j P(X\geq j) + (a_1 + \dots + a_{n+1})P(X = n+1) - P(X = n+1) \sum_{j=1}^n a_j
\\&+ P(X \geq n+2) a_{n+1}
\\=&\sum_{j=1}^n a_j P(X\geq j) + a_{n+1}P(X = n+1) + P(X \geq n+2) a_{n+1}
\\=&\sum_{j=1}^n a_j P(X\geq j) + a_{n+1}P(X \geq n+1)
\\=&\sum_{j=1}^{n+1} a_j P(X \geq j).
\end{align}$$
A: Here is a direct, graphical, proof.
Let $p_k:=P(X=k)$ for $k=1\cdots n$ and $p_+:=P(X\geq n+1).$
We have clearly $(\sum_{k=1}^n p_k)+p_+=1.$
Have a look now at the graphics below where the same stairlike figure is represented twice, divided 


*

*on the left into horizontal stripes, with areas $(a_1+a_2+ \cdots + a_j) \times p_j$ and $(a_1+a_2+ \cdots + a_n) \times p_+$ for the last one. This represents the LHS of the formula to be established.

*on the right into vertical stripes, with areas  $(p_j+p_{j+1}+ \cdots p_n+p_+) \times a_j$, i.e., $P(X \geq j) \times a_j$. This is for the RHS of the formula.

