Assuming that a continuous random variable X has a density denoted by $f_X$. Show that if the k-th moment of X exists (for k ∈ N+), that is E(|X|$^k$) < ∞, then E(|X|$^s$) < ∞ for all 0 < s < k.
It intuitively makes sense for a lower moment to exist knowing a higher moment exists, but nevertheless I have got no idea on how to show that. Please help.
Hint: Use Jensen's inequality.