No dominating function exists for the “moving bumps”

One can easily apply the dominated convergence theorem (DCT) to see that there is no integrable dominating function for each of the following functions

• $f_n=1_{[n,n+1]}$,
• $h_n=\frac{1}{n}1_{[0,n]}$,
• $g_n=n1_{[1/n,2/n]}$.

For "integrable dominating function" I mean the dominating function in the assumption of the DCT. I would like to see why in each of these examples, no dominating function exists without appealing to the DCT.

If $G\geq f_n$ for all $n$, then $G\geq 1$ a.e..

If $G\geq h_n$ for all $n$, then $$\int G\geq \sum_{n=1}^\infty\frac{1}{n}.$$

Could anyone help me with the third one?

For each $n$, $$g_n(x):= nI_{\left[\frac1n, \frac2n\right]}(x)\ge \frac1x I_{\left[\frac1n,\frac2n\right]}(x)\ge \frac1xI_{\left(\frac2{n+1},\frac2n\right]}(x).$$ Therefore if $G(x)\ge g_n(x)$ for all $n$, then $G(x)\ge\frac1x$ on the interval $(0,2]$.
If $\forall n,\ G\ge g_n$, then $\forall n,\ G\ge g_{2^n}$. Since $[2^{-n},2^{-n+1}]\cap [2^{-m},2^{-m+1}]$ is negligible whenever $m\ne n$, you get that $$\sup_{n\in\mathbb N} g_{2^n}=\sum_{k=1}^\infty g_{2^k}$$ hence
$$G\ge \sum_{k=1}^\infty g_{2^k}$$
But then $$\int G\,dx\ge \sum_{k=1}^\infty 2^k\cdot 2^{-k}=+\infty$$