I am reading D. S. Jones' book, The theory of generalised functions, and this question is specifically about theorem 3.16 on page 81. To make this post self-consistent I'll give a bit of context. Let's start with the definitions.
A good function $\gamma(x)$ is defined as a function acting on reals, infinitely differentiable and such that it and all its derivatives are a $O\left(|x|^{-N}\right)$, for all $N$.
A generalized function is by definition a family of good functions $\{\gamma_n(x)\}_{n\in\mathbb{N}}$ such that the limit exists $\lim_{n\to\infty}\int_{-\infty}^\infty\gamma_n(x)\gamma(x)\mathrm{d}x$ for all good $\gamma(x)$.
I also report a lemma, without proving it.
The following property is valid for all good functions: $$\max_{x\in\mathbb{R}}\left\{(1+x^2)^{\alpha} \left|\gamma^{(p)}(x)\right| \right\}\leq \left(\frac{\pi}{2}\right)^n\max_{x\in\mathbb{R}}\left\{(1+x^2)^{\alpha+n} \left|\gamma^{(p+n)}(x)\right|\right\}$$ where $\alpha\geq 0$ and $p\in\mathbb{N}_0$.
Having said that, let's get to the doubt. Theorem 3.16 states that:
Whatever the generalized function is $ \left\{\gamma_n(x)\right\}_{n\in\mathbb{N}} $, there exist $k$, $r$, and a constant $K$ such that $$ \left| \lim_{n\to\infty}\int_{-\infty}^\infty \gamma_n(x) \gamma(x) \mathrm{d}x\right| \leq K \max_{x\in\mathbb{R}}(1+x^2)^{k/2}\gamma^{(r)}(x)$$ for all good $\gamma$.
He begins the proof by noting that:
$$\begin{split}\left| \int_{-\infty}^\infty \gamma_n(x) \gamma(x) \mathrm{d}x\right| &\leq \int_{-\infty}^\infty |\gamma_n(x)| \mathrm{d}x \cdot \max_{x\in\mathbb{R}}| \gamma(x)|\\ &\underbrace{\leq}_{\text{Lemma}} \underbrace{ \left(\frac{\pi}{2}\right)^s \int_{-\infty}^\infty |\gamma_n(x)| \mathrm{d}x}_{C_{s,n} \text{ indipendent of }\gamma} \cdot \max_{x\in\mathbb{R}}(1+x^2)^{s}| \gamma^{(s)}(x)|\end{split}$$
and he says that the proof would be concluded if for some $s$ the constants $C_{s,n}$ would be bounded as $n$ varies. In order to prove this, he proceeds by contradiction, assuming that this is not true, and so equivalently that chosen any $k$ and $s$, you will always find an index $n$ beyond which $ \left| \int_{-\infty}^\infty \gamma_n(x) \gamma(x) \mathrm{d}x\right| > L_n\cdot \max_{x\in\mathbb{R}}(1+x^2)^{k/2}| \gamma^{(s)}(x)| $, no matter how $L_n$ is large.
It is precisely the passage "and so equivalently that" that I dispute. For me, the absurd hypothesis that could be set are at most two and I didn't quite understand which of the two he had in mind to follow, from what he wrote:
- I assume by contradiction that, whatever $s$ is, the sequence $C_{s,n}$ is always unbounded;
- I suppose absurdly that, however they are chosen $k,r$ and $K$, you can always find a good function $\tilde{\gamma}$ such that the sequence $ \left| \int_{-\infty}^\infty \gamma_n(x) \tilde{\gamma}(x) \mathrm{d}x\right| $ is not bounded as $n$ varies.
By choosing the first hypothesis, it does not seem to me that much can be deduced, because if the upper bounds are unlimited it does not mean that the sequence must also be unlimited. The second hypothesis by contradiction instead allows free choice for the constants $k,r$ and $K$ (those of the statement of the theorem), but then the good function $\tilde{\gamma}(x)$ is "the one that comes out", but, instead, he bases the entire continuation of the proof on the possibility of choosing the good functions as he likes, in the way that is most convenient for him to reach a contradiction.
What am I missing? I can't believe that the passage highlighted by me is truly a logical error committed by the author. Much more likely I'm missing something and I ask for help in understanding this.
Thank you.