Alternative Expected Value Proof

I am currently tasked with proving an alternative definition of the expected value function.

Considering X to be a random variable that takes all positive integers, I have to prove that
$E[X]=\sum\limits_{i=1}^{\infty} P[X\ge i]$.

So far I’ve gotten to the fact that $P[X\ge i] = \sum\limits_{j=i}^{\infty} P[X=j]$ however I just have no idea where to go from here. I have no idea how that relates to expected value in any way.

I’m most certainly not looking for the answer, but a friendly nudge in the right direction would be extremely helpful.

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