Let $\;u,v:\mathbb R \to \mathbb R^n\;$ be two continuous functions and consider the sequence $\;f_n(x):={\vert u(x)-v(x-y_n) \vert}^2\;$ where $\; \vert \cdot \vert \;$ stands for the Euclidean norm and $\;y_n\in \mathbb R\;$.

I know that $\;v(x-y_n) \to a \in \mathbb R^n\;$ as $\;y_n \to \infty\;$ and hence $\;f_n(x):={\vert u(x)-v(x-y_n) \vert}^2 \to {\vert u(x)-a \vert}^2\;$ as $\;n\to \infty\;$

I need a bound from below for $\;\int_{\mathbb R} f_n(x)\;dx=\int_{\mathbb R} {\vert u(x)-v(x-y_n) \vert}^2 \;dx\;$ and this is why Fatou's lemma came to my mind.


Is it true to claim (EDITED CLAIM) that:

$\;\int_{\mathbb R} f_n(x)\;dx=\int_{\mathbb R} {\vert u(x)-v(x-y_n) \vert}^2 \;dx \ge \int_{I} {\vert u(x)-v(x-y_n) \vert}^2 \;dx \gt\\ \liminf \int_{I} {\vert u(x)-v(x-y_k) \vert}^2 -\varepsilon\;dx\;\ge \int_{I} {\vert u(x)-a \vert}^2\;dx-\varepsilon\;$

$\;\forall \varepsilon \gt 0\;$ and $\;\forall k \ge K_0\;$ Note: $\;I\;$ is an open interval of $\;\mathbb R\;$

The reason I doubt about the above inequalities is firstly because I'm not sure if $\;f_n\;$ is a sequence of measurable functions (I think it should be, since $\;u,v\;$ are continuous) and secondly I don't know if the part of strictly inequality holds.

Any help here would be valuable!

Thanks in advance!


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