I am looking at the proof of the logarithmic Sobolev inequality (theorem 8.14) in the book Analysis of Lieb--Loss:
Suppose that $f \in H^1(\mathbb{R}^n)$ and let $a>0$ be any number. Then
$$\dfrac{a^2}{\pi}\int_{\mathbb{R}^n}\lvert \nabla f(x) \rvert^2 \geq\int_{\mathbb{R}^n}\lvert f(x) \rvert^2 \ln \bigg(\dfrac{\lvert f(x) \rvert^2}{\lVert f \rVert_2^2} \bigg) dx + n(1+ \ln a)\lVert f \rVert_2^2.$$
I understand basically everything up to the approximation argument in the end. So we have proved this holds for $f \in H^1(\mathbb{R^n})\cap L^{2-\delta}(\mathbb{R}^n) \cap L^{2+\delta}(\mathbb{R}^n)$. Then they claim that the general inequality follows from a "standard approximation argument". Does this follow from the fact that
$$\overline{L^{2-\delta}(\mathbb{R}^n) \cap L^{2+\delta}(\mathbb{R}^n)} = L^2(\mathbb{R}^n)?$$
If it does, then I tried to use this explicitly and it doesn't really work out for me, but I can't see what I'm doing wrong. I also don't get the fact that for $\varepsilon > 0$, we get that $\ln \lvert f(x) \rvert^2 < \lvert f(x)\rvert^{\varepsilon}$ and that this, together with Sobolev's inequality gives us that $$\int_{\mathbb{R}^n}\lvert f(x)\rvert^2(\ln \lvert f \rvert^2)_+$$ is well defined and finite. This is all good of course, but how does this help us?
If you have any thoughts at all about this, I would be happy to hear them!