Is the following statementt is True/false ??
Let $ \{f_n\}$ be a sequence of continuous real valued functions defined on $\mathbb{R}$ converging uniformly on $\mathbb{R}$ to a function $f$.
If each of the functions $f_n$ is integrable, then
$$\lim_{n\to \infty} \int_\infty^{-\infty} f_n(t)\,dt=\int_\infty^{-\infty } f(t) \, dt$$
MY attempt :
It will True by Lebesgue dominated convergence theorem and another approach is that i thinks it will also be True because if i take $f_n(t) = \frac{|sint|} {n}$ it will satisfied this statement
Best Answer
It's not true. Try a very short but even longer triangle sitting on the $x$ axis. Bad picture:
$$ \_\Delta\_$$ eg $1/n$ height and $n^{2}$ width. The convergence to 0 is uniform but the integrals get larger.
Explicit formula: define $T(x)=\max(0,1-|x|)$. Then set $$f_n(x)=\frac{1}{n} T\left(\frac x{n^{2}}\right)$$
I suggest you draw these, but use your highschool formula for the area of a triangle instead of trying to compute the integral.