Let $(X,M, μ)$ be a measure space and $0 < p < q ≤ ∞$. Prove: $L^p(X)$ is not contained in $L^q(X)$ iff $X$ contains sets of arbitrarily small positive measure.
My work:
I proved the forward direction. For backward direction, considering $q<\infty$, if I take disjoint sets $E_n\subset X$ such that $0<\mu(E_n)<\frac{1}{2^n}$, then for $f=\sum_{n=1}^\infty \mu(E_n)^{(-1/q)}\chi_{E_n}$, I can show that $f\in L^p$ and $f\notin L^q$.
Now for the case $q=\infty$ how can I find a function $g$ so that $g\in L^p$ and $g\notin L^\infty$. Can anybody please give me a hint?
EDIT: Does $g=\sum_{n=1}^\infty \chi_{E_n}$ will work? I can prove that $g\in L^p$. But struggling to prove that $g\notin L^\infty$. Any ideas?