Let $(X,\mathcal{A},\mu)$ be a measure space, $p\in{}[1,+\infty)$ and $(f_n)_{n\in\mathbb{N}}$ a sequence of function in $\mathcal{L}^p(X,\mathcal{A},\mu)$. Show that there exists an $F\in\mathcal{L}^p(X,\mathcal{A},\mu)$ and subsequence $(f_{k_n})_{n\in\mathbb{N}}$ such that $f_{k_n}\leq{}|F|,\forall{}n\in\mathbb{N}$.
I am meant to prove this but confused about why it even holds: for example considering the function $f_n(x)=n\mathbb{1}_{[0,\frac{1}{n})}(x)$ isn't $f_n$ a clear counterexample?
Any suggestions?
What you want to prove is simply wrong.
To see this, take your favorite function $f \neq 0$ with $f \in L^p$, and define $f_n := n \cdot f$. Clearly, no function as you want can exist.
Even if you assume your sequence to be bounded, your claim is not true. To see this, consider $f_n = 1_{[n,n+1]}$ with $(X,\mu)$ the real line equipped with the Lebesgue measure.