Q:

DCT does not hold ( Explanation of another answer)

The question is here Find an example given Hypotheses , DCT does not hold 
I would really appreciate it if you could clarify the given answer to the above question.
Can the answer be written as the following: ( I is indicator function)  
Given a measure space $( \Bbb{R} , \mathcal{R} , λ)$ and $h_n(ω)= n I_{[2^n,2^n+\frac{1}{n^2}]}$ Then $0  \leq h_n(ω)=n I_{[2^n,2^n+\frac{1}{n^2}]}  (ω) \to 0  for all ω \in \Bbb{R}$. and $h \geq \int h_n dλ = \int \sum_{n=1}^\infty n I_{[2^n,2^n+\frac{1}{n^2}]} dλ$ ??
Also, why is true that $h\geq \sum_{n=1}^\infty n E_{[2^n,2^n+\frac{1}{n^2}]}$
what is h equal to 
If that is wrong, how can we replace the characteristic function given in the answer  $f_n=g_n=h_n=nE_{[2^n,2^n+\frac{1}{n^2}]}(x)$
I am sorry if the answer is obvious/trivial. 

A:

Because the intervals $[2^{-n},2^{-n}+n^{-2}]$ are all disjoint then if such $h\geqslant |f_n|$ for all $n\in\Bbb N$ would exists then $h\geqslant |f_1|$ and $h\geqslant |f_2|$ so, $h\geqslant |f_1|+|f_2|$, etc...

