Let $\;u,v:\mathbb R \to \mathbb R^m\;$ be two functions such that: $\; \vert u(x)-l_{+} \vert \le e^{-kx}\;$,$\; \vert u(x)-l_{-} \vert \le e^{kx}\;;$ as $\ x \to +\infty\;$ and $\; x \to -\infty\;$ respectively. Similarly for $\;v\;$. (NOTE: $\;k \gt 0\;$ and $\;l_{-} \neq l_{+}\;$)
If $\;y_n \in \mathbb R\;$ is a unbounded sequence, in the sense that $\;y_n \to \infty\;$ as $\;n \to \infty\;$, and $\;{\vert u(x)-v(x-y_n) \vert}^2 \to {\vert u(x) - l_{-} \vert}^2\;$ as $\;n \to \infty\;$, prove that $\;\int_{\mathbb R} {\vert u(x)-v(x-y_n) \vert}^2 \;dx\to \infty\;$ as $\;n \to \infty\;$
I observed that $\;\int_{\mathbb R} {\vert u(x)-l_{-} \vert}^2 \;dx\;=\infty\;$ but I can't see how to take advantage of this fact in order to show the $\;L^2-$norm of $\;u(x)-v(x-y_n) \;$ goes also to $\;\infty\;$.
Am I missing some crucial Theorem here?
I 'm having a really hard time getting my head around this so any help or even hints would be valuable!
Thanks in advance!
Let $f_n =|u(x)-v(x-y_n)|^{2}$ and $f_n =|u(x)-l_{-}|^{2}$. You have stated that $f_n \to f$ a.e. and $\int f= \infty$. By Fatou's Lemma $\infty = \int f = \int \liminf f_n \leq \liminf \int f_n$ so $\int f_n \to \infty$.