existence of moments question

I was ready the book **"****Introduction to probability theory" by James R. ****McCord**, III [and] Richard M. **Moroney**, Jr and it gave this thereom about the existence of moments: if the mth raw moment of a random variable X exists, then the Xth and all preceding moments of X about any finite point all exist.

It then said that this follows from simple computations and the relation:

E(|X|^(m-1))≤|X|^(m + 1)

Which is true because: |x|^(m-1)≤|x|^(m+1).

For all real x and all m≥1.

I am not understanding any of this at all!!! Can someone help explain to me what is going on?