Currently I'm reading "Model Completeness Results for Expansions of the Ordered Field of Real Numbers by Retricted Pfaffian Functions and the Exponential Function" by Wilkie and I do not fully understand the proof of Claim 8.4 (and 8.5) on page 1081. In particular how the result 3.5 (page 1062) was applied.

Corollary 3.5
Suppose $\widetilde{K} \models \widetilde{T}$, $e \in K$ and $g:(e,\infty) \rightarrow K$ is a $\widetilde{K}$ definable function which is not eventually identically zero. Then there is a rational number $s$ and a non-zero element $a \in K$ such that $g(x)x^s \rightarrow a$ as $x \rightarrow \infty$ (in the sense of $\widetilde{K}$).

Note that $\widetilde{T}$ is the theory $\text{Th}(\mathbb{R}_\text{Pfaff})$.

I have difficulties with the following applications of Corollary 3.5 in Claim 8.4 (page 1081):

It clearly follows from 3.5 (with $\widetilde{K} = \widetilde{k}$, $g = \psi_i$) that $\psi_i(t) \rightarrow a_i$ as $t \rightarrow \infty$ for some $a_i \in k$ [...].

Further, by applying 3.5 again with $g = \psi_i-a_i$, there exists a positive rational, $\theta_i$ say, such that $|\psi_i(t)-a_i| < t^{-\theta_i}$ for all sufficiently large $t \in k$.

Exact details can be found in Wilkie's paper https://www.ams.org/journals/jams/1996-9-04/S0894-0347-96-00216-0/S0894-0347-96-00216-0.pdf on page 1062 and 1081.

How are those two statements derived using Corollary 3.5? I feel like I have overlooked some simple arguments but I do not know why the rational powers vanished or how the inequality is justified. (Also, the paper works with restricted Pfaffian functions and thus restricted domains which let bounded variables only range over the interval $(0,1)$. I hope that the above in particular does not need any arguments that refer to this restrictedness.)

I'm grateful for every help.

Currently I'm reading "Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function" by Wilkie, and I do not fully understand the proof of Claim 8.4 (and 8.5) on page 1081. In particular how the result 3.5 (page 1062) was applied.

Corollary 3.5
Suppose $\widetilde{K} \models \widetilde{T}$, $e \in K$ and $g:(e,\infty) \rightarrow K$ is a $\widetilde{K}$ definable function which is not eventually identically zero. Then there is a rational number $s$ and a non-zero element $a \in K$ such that $g(x)x^s \rightarrow a$ as $x \rightarrow \infty$ (in the sense of $\widetilde{K}$).

Note that $\widetilde{T}$ is the theory $\text{Th}(\mathbb{R}_\text{Pfaff})$.

I have difficulties with the following applications of Corollary 3.5 in Claim 8.4 (page 1081):

It clearly follows from 3.5 (with $\widetilde{K} = \widetilde{k}$, $g = \psi_i$) that $\psi_i(t) \rightarrow a_i$ as $t \rightarrow \infty$ for some $a_i \in k$ [...].

Further, by applying 3.5 again with $g = \psi_i-a_i$, there exists a positive rational, $\theta_i$ say, such that $|\psi_i(t)-a_i| < t^{-\theta_i}$ for all sufficiently large $t \in k$.

Exact details can be found in Wilkie's paper (link at AMS site) on page 1062 and 1081.

How are those two statements derived using Corollary 3.5? I feel like I have overlooked some simple arguments but I do not know why the rational powers vanished or how the inequality is justified. (Also, the paper works with restricted Pfaffian functions and thus restricted domains which let bounded variables only range over the interval $(0,1)$. I hope that the above in particular does not need any arguments that refer to this restrictedness.)

I'm grateful for every help.

Currently I'm reading "Model Completeness Results for Expansions of the Ordered Field of Real Numbers by Retricted Pfaffian Functions and the Exponential Function" by Wilkie and I do not fully understand the proof of Claim 8.4 and 8.5 on page 1081. In particular how the result 3.5 (page 1062) was applied.

Corollary 3.5
Let $K \models \text{Th}(\mathbb{R}_\text{Pfaff})$ and let $e \in K$. Further, let $g: (e,\infty) \rightarrow K$ be a definable function of $K$ which is not eventually identically zero. Then there is some $s \in \mathbb{Q}$ and some $a \in K \setminus \lbrace 0 \rbrace$ such that $g(x)x^s \rightarrow a$ as $x \rightarrow \infty$ (with respect to K).

As Claim 8.5 is shown in the same way as Claim 8.4, we will just look at the application of 3.5 in Claim 8.4:
For $K = k$ and $g = \psi_i$ a direct application of 3.5 yields that $\psi_i \rightarrow a_i$ as $t \rightarrow \infty$ for some $a_i \in k$. And another application of 3.5 for $g = \psi_i-a_i$ yields that there is a positive rational $\theta_i$ such that $|\psi_i(t)-a_i| < t^{-\theta_i}$ for all sufficiently large $t \in k$. Exact details can be found in Wilkie's paper https://www.ams.org/journals/jams/1996-9-04/S0894-0347-96-00216-0/S0894-0347-96-00216-0.pdf on page 1062 and 1081.

I feel like I have overlooked some simple arguments but I do not know why the rational powers vanished or how the inequality is justified. (Also, the paper works with restricted Pfaffian functions and thus restricted domains which let bounded variables only range over the interval $(0,1)$. I hope that the above in particular does not need any arguments that refer to this restrictedness.)

I'm grateful for every help.

Currently I'm reading "Model Completeness Results for Expansions of the Ordered Field of Real Numbers by Retricted Pfaffian Functions and the Exponential Function" by Wilkie and I do not fully understand the proof of Claim 8.4 (and 8.5) on page 1081. In particular how the result 3.5 (page 1062) was applied.

Corollary 3.5
Suppose $\widetilde{K} \models \widetilde{T}$, $e \in K$ and $g:(e,\infty) \rightarrow K$ is a $\widetilde{K}$ definable function which is not eventually identically zero. Then there is a rational number $s$ and a non-zero element $a \in K$ such that $g(x)x^s \rightarrow a$ as $x \rightarrow \infty$ (in the sense of $\widetilde{K}$).

Note that $\widetilde{T}$ is the theory $\text{Th}(\mathbb{R}_\text{Pfaff})$.

I have difficulties with the following applications of Corollary 3.5 in Claim 8.4 (page 1081):

It clearly follows from 3.5 (with $\widetilde{K} = \widetilde{k}$, $g = \psi_i$) that $\psi_i(t) \rightarrow a_i$ as $t \rightarrow \infty$ for some $a_i \in k$ [...].

Further, by applying 3.5 again with $g = \psi_i-a_i$, there exists a positive rational, $\theta_i$ say, such that $|\psi_i(t)-a_i| < t^{-\theta_i}$ for all sufficiently large $t \in k$.

Exact details can be found in Wilkie's paper https://www.ams.org/journals/jams/1996-9-04/S0894-0347-96-00216-0/S0894-0347-96-00216-0.pdf on page 1062 and 1081.

How are those two statements derived using Corollary 3.5? I feel like I have overlooked some simple arguments but I do not know why the rational powers vanished or how the inequality is justified. (Also, the paper works with restricted Pfaffian functions and thus restricted domains which let bounded variables only range over the interval $(0,1)$. I hope that the above in particular does not need any arguments that refer to this restrictedness.)

I'm grateful for every help.