$L^\infty$ and the intersection of the spaces $L^p$ I'm trying to understand if it's true that:
" if $f\in L^p\quad \forall p\in N\implies f\in L^\infty$"?
My thoughts:
Since $\int_R |f(x)|^p dx<\infty\quad\forall p\implies |f(x)|\to 0$?
Can anyone help me please? Thank you.
 A: Hint: Consider
$$
f(x)=\left\{\begin{array}{}
-\log(x)&\text{if }x\in(0,1)\\
0&\text{otherwise}
\end{array}\right.
$$
A: Consider the function f in $R$ constructed as follows:
Let M is a big natural number. Then let $m${x $\in R$:f(x)=M}= $\frac{1}{M}$.
(where m is the lebesgue measure in $R$). 
 Then $m${x $\in R$:f(x)=M+1} =$\frac{1}{(M+1)^2}$. Carry on this type of construction i.e in general $m${x $\in R$:f(x)=M+k} =$\frac{1}{(M+k)^{k+1}}$. 
Let {x $\in R$:f(x)=M+k}=$E_k$ $\forall k \in N$. Let $E_n$'s be such that each $E_n$ is an interval and they are mutually disjoint.(This type of construction is possible since $\sum (\frac{1}{M}+ \frac {1}{M+1^2} + ...)$ $< \infty$).
And also f(x)=0 elsewhere. Now it is clear that f $\notin L^\infty$ and $|f(x)| \longrightarrow 0$ is not true. It is clear that f is a measurable function.We show that $f \in L^p \forall p \in N$. For some $k \in N$, $\int_{R}f^k dm$= $M^{k-1}+(M+1)^{k-2}+...+1+ \sum (\frac{1}{M+k}+\frac{1}{(M+k+1)^2}+...+\frac{1}{(M+k+n)^{n+1}}+...) < \infty$. Hence $f \in L^k \forall k \in N$. 
