Sequence with fixed index that dominate

Question

Suppose I have a sequence $a_n$, $b_n$ of real numbers, and sequence $f_n$ and $g_n$ for functions. I know that $a_n\to a\neq 0$ and $$ \lim_{n\to\infty} \int| a_n f_n(x)+b_n g_n(x)|dx =M<+\infty $$ My question: can I find a interger $N_0\in\mathbb N$ large such that, up to subsequence of $f_n$, $b_n$, $g_n$, $$ \sup_{n\geq N_0}\int| a_{N_0} f_n(x)+b_n g_n(x)|dx<+\infty? $$

Answer 1

Suppose that $f_n=1=g_n$, i.e., the constant function. Suppose that $b_n=-a_n$. In this case, $$ \int|a_nf_n(x)+b_ng_n(x)|dx=\int 0dx=0. $$ However, if $b_m\not=-a_n$, then $$ \int|a_mf_n(x)+b_ng_n(x)|dx=\infty. $$ This can be generalized (using the $\sup$ in the question) to apply when $f_n\in L^1$ for all $n$ - for example, over $\mathbb{R}$, let $f_n=1_{[0,n]}=g_n$ be the indicator function).

For your particular property, observe that \begin{align*} \int|a_{N}f_n(x)+b_ng_n(x)|dx&=\int|a_{N}f_n(x)-a_nf_n(x)+a_nf_n(x)+b_ng_n(x)|dx\\ &\leq\int|a_{N}f_n(x)-a_nf_n(x)|dx+\int|a_nf_n(x)+b_ng_n(x)|dx\\ &\leq |a_N-a_n|\int|f_n(x)|dx+\int|a_nf_n(x)+b_ng_n(x)|dx. \end{align*} Therefore, what is sufficient for your condition is that $\int|f_n(x)|dx$ is uniformly bounded in $n$.