Series of functions: convergence interval type vs. monotony of the function and calculation of supx∈[r,∞)|fn(x)|\mathrm{sup}_{x \in [r,\infty)}|f_n(x)|

I have two doubts regarding the normal convergence of a function series.

Consider a real functions series fn(x):A⊂R→Rf_n(x): A \subset \mathbb{R} \to \mathbb{R} of wich I want to study the normal convergence.
∑n≥0fn(x)\sum_{n \geq 0} f_n(x)

Suppose that I find that the series
∑n≥0|fn(x)|\sum_{n \geq 0} |f_n(x)|
converges pointwise in a open interval I=(a,+∞)I=(a,+ \infty).

Since normal convergence is stronger than pointwise convergence of ∑n≥0|fn(x)|\sum_{n \geq 0} |f_n(x)|, only x∈Ix \in I should be considered. Besides that, proving that series converges in a open interval (as II is) would imply that it converges normally also in the corresponding closed interval, that is I′=[a,∞)I’=[a,\infty), but this would also imply that the series ∑n≥0|fn(x)|\sum_{n \geq 0} |f_n(x)| converges pointwise in x=ax=a which is not true. Therefore I will look in “smaller” intervals inside II, so for x∈Jr=[r,∞)x \in J_r=[r,\infty) with r>ar>a.

The normal convergence requires to find some constants MnM_n such that