Finding the expected value of a nonnegative integer valued random variable If $X$ is a nonnegative integer valued random variable, show that

$$\mathbb{E}(X) = \sum_{i=1}^\infty\mathbb{P}(X\geq i) = \sum_{i=0}^\infty \mathbb{P}(X\geq i).
$$

I'm not sure how to do this. I only know the definition that
$$\mathbb{E}(X):= \sum_{i=-\infty}^\infty i \mathbb{P}(X=i)\mathbf{1}_{\{X\geq 0\}} = \sum_{i=1}^\infty i \mathbb{P}(X=i).
$$
Assert that most surely $X\in \Bbb N^+$ , then use that for all $k\in\Bbb N^+$, by definition of counting: $k=\sum_{j=0}^{k-1} 1$ .

$$\mathsf E(X) {= \sum_{k=1}^\infty k~\mathsf P(X=k) \\ = \sum_{k=1}^\infty\mathsf P(X=k)\sum_{j=0}^{k-1}1\\ \vdots \\ = \sum_{j=0}^\infty \mathsf P(X&gt;j)}$$
If $E[X] &lt; \infty$, then see here: 


Expectation of an non-negative integer-valued random variable
Find the Mean for Non-Negative Integer-Valued Random Variable


If $E[X] = \infty$, I guess you have to show

$$\lim_{i \to \infty} P(X \ge i) \ne 0 \leftarrow \lim_{i \to \infty} iP(X = i) \ne 0$$