Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$ I aim to show that $\int_{(0,1]} 1/x = \infty$.  My original idea was to find a sequence of simple functions $\{ \phi_n \}$ s.t $\lim\limits_{n \rightarrow \infty}\int \phi_n = \infty$.  Here is a failed attempt at finding such a sequence of $\phi_n$:
(1) Let $A_k = \{x \in (0,1] : 1/x \ge k  \}$ for $k \in \mathbb{N}$.
(2) Let $\phi_n = n \cdot \chi_{A_n}$
(3) $\int \phi_n = n \cdot m(A_n) = n \cdot 1/n = 1$
Any advice from here on this approach or another?

Write $I_k:=((k+1)^{-1},k^{—1})$. Then for each $n$, 
$s_n:=\sum_{k=1}^nk\chi_{I_k}$ is a simple non-negative function, and $0\leq s_n\leq f(x):=1/x$. We have 
$$\int_{(0,1]}s_n \, d\lambda=\sum_{k=1}^nk\left(\frac 1k-\frac 1{k+1}\right)=\sum_{k=1}^nk\frac{k+1-k}{k(k+1)}=\sum_{k=1}^n\frac 1{k+1}.$$
So 
$$\int_{(0,1]}s_{2n} \, d\lambda-\int_{(0,1]}s_n \, d\lambda=\sum_{k=n+1}^{2n}\frac 1{k+1}\geq\frac n{2n+1}\geq \frac 13.$$
As the sequence $\{\int_{(0,1]}s_n \, d\lambda\}$ is increasing, it has a limit. This one can't be finite by the last inequality, and the sequence is non-negative, so it converges to $+\infty$. This proves that 
$$\sup\{\int_{(0,1]}s \, d\lambda,0\leq s\leq f, s\text{ simple}\}$$
is infinite, that is, $f$ is not Lebesgue integrable. 
