Question on whether an integral is finite or infinite using certain inequality

Question

Let $\;f_n :\mathbb R \to \mathbb R^m\;$ be a sequence of $\;L^2-$functions and suppose that $\;\exists N_0\;$ such that

$\;\forall \varepsilon \gt 0\;,\;\forall n \ge N_0\;,:\;\int_{I} {\vert f_n(x) \vert}^2\;dx\; \gt \int_{I} {\vert g(x) \vert}^2\;dx\;-\varepsilon\;\;\;(*)$

where $\;I\subset \mathbb R\;$ and $\;g:\mathbb R \to \mathbb R^m\;$ a smooth function.

QUESTION:

If I know that when $\;I \to \mathbb R\;\Rightarrow \int_{I} {\vert g(x) \vert}^2\;dx \to \infty\;$ then is it true to deduce from $\;(*)\;$ that $\;\int_{\mathbb R} {\vert f_n(x) \vert}^2\;dx=\infty\;$?

I'm having a really hard time understanding how inequalities such $\;(*)\;$ work so any help would be valuable!

Thanks in advance.

Answer 1

No. Let $g(x)=1$ for all $x\in \mathbb{R}$, and $f_n(x)=e^{-(x/n)^2}$. Then on a bounded interval $I$, we have $f_n\to 1=g$ uniformly, therefore $\|f_n\|_{L^2(I)}^2\to \|g\|_{L^2(I)}^2$.

However, $\|g\|_{L^2(\mathbb{R})}= \infty,$ while $\|f_n\|_{L^2(\mathbb{R})}<\infty$ for all $n$.