We can write, since $f\in\mathcal S(\mathbb R^n)$
\begin{align*}
\lVert x^{\alpha}\partial^{\beta}f\rVert_1&\leqslant
\int_{\mathbb R^n}|x|^{\alpha}|\partial^{\beta}f(x)|dx\\\
&=\int_{\mathbb R^n}(1+|x|)^{n+1}|x|^{\alpha}|\partial^{\beta}f(x)|\frac 1{(1+|x|)^{n+1}}dx\\\
&\leqslant C'\sup_{x\in\mathbb R^n}(1+|x|)^{n+1}|x|^{\alpha}|\partial^{\beta}f(x)|
\int_{\mathbb R^n}\frac{dx}{(1+|x|)^{n+1}}\\\
&=C'\sup_{x\in\mathbb R^n}(1+|x|)^{n+1}|x|^{\alpha}|\partial^{\beta}f(x)|
s_n\int_0^{+\infty}\frac{r^{n-1}}{(1+r)^{n+1}}dr,
\end{align*}
where $s_n$ is the area of the unit sphere in $\mathbb R^n$. The last integral
is convergent, and we get the expected result putting $C:=C's_n\int_0^{+\infty}\frac{r^{n-1}}{(1+r)^{n+1}}dr$.
For the second fact, note that $\partial^{\beta}\widehat f(x)=\int_{\mathbb R^n}
i^{\beta}t^{\beta}e^{it\cdot x}f(t)dt$, hence for $x\in\mathbb R^n$:
\begin{align*}
(1+|x|)^N|\partial^{\beta}\widehat f(x)|&=
(1+|x|)^N\left|\int_{\mathbb R^n}e^{it\cdot x}t^{\beta}f(t)dt\right|\\\
&=\sum_{k=0}^N\binom Nk|x|^k\left|\int_{\mathbb R^n}e^{it\cdot x}
t^{\beta}f(t)dt\right|\\\
&=\sum_{k=0}^N\binom Nk\sum_{|\gamma |=k}\left|\int_{\mathbb R^n}
x^{\gamma}e^{it\cdot x}
t^{\beta}f(t)dt\right|\\\
&=\sum_{|\gamma|\leqslant N}\binom Nk\left|\int_{\mathbb R^n}
x^{\gamma}e^{it\cdot x}t^{\beta}f(t)dt\right|.
\end{align*}
Let $\displaystyle I_{\gamma}(x):=\int_{\mathbb R^n}
x^{\gamma}e^{it\cdot x}t^{\beta}f(t)dt$. Integrating by parts and using
Leibniz formula, we have
\begin{align*}
|I_{\gamma}(x)|&=\left|\int_{\mathbb R^n}e^{it\cdot x}\sum_{\alpha\leqslant
\gamma}\binom{\gamma}{\alpha}\partial^{\alpha}f(t)t^{\beta-\alpha}\frac{\beta !}{(\beta-\alpha)!}dt\right|\\\
&\leqslant \beta !\sum_{\alpha\leq \gamma}\frac 1{(\beta-\alpha)!}\binom{\gamma}{\alpha}\int_{\mathbb R^n}\left|\partial^{\alpha}f(t)t^{\beta-\alpha}\right|dt,
\end{align*}
and using the first point
\begin{align*}
|I_{\gamma}(x)|&\leqslant \beta !\sum_{\alpha\leqslant \gamma}\frac 1{(\beta-\alpha)!}\binom{\gamma}{\alpha}C_{\alpha}\sup_{x\in\mathbb R^n}
(1+|x|)^{n+1}|x|^{\beta-\alpha}|\partial^{\alpha}f(x)|\\\
&\leqslant \beta !\sum_{\alpha\leqslant \gamma}\frac 1{(\beta-\alpha)!}\binom{\gamma}{\alpha}C_{\alpha}\sup_{x\in\mathbb R^n}
(1+|x|)^{n+1}(1+|x|)^{\beta-\alpha}|\partial^{\alpha}f(x)|\\\
&\leqslant \beta !\sum_{\alpha\leqslant \gamma}\frac 1{(\beta-\alpha)!}\binom{\gamma}{\alpha}C_{\alpha}\sup_{x\in\mathbb R^n}
(1+|x|)^{n+1+\beta}|\partial^{\alpha}f(x)|\\\
&\leqslant \beta !\sum_{\alpha\leqslant \gamma}\frac 1{(\beta-\alpha)!}\binom{\gamma}{\alpha}C_{\alpha}\lVert f\rVert_{(n+1+\beta,\alpha)}.
\end{align*}
Putting $A_{\gamma,\beta}=\beta\max_{\alpha\leqslant \gamma}\frac 1{(\beta-\alpha)!}\binom{\gamma}{\alpha}C_{\alpha}$. Then
$|I_{\gamma}(x)|\leqslant A_{\gamma,\beta}\sum_{\alpha\leq\gamma}\lVert f\rVert_{(n+1+\beta,\alpha)}$. Now, put $\displaystyle B_{N,\beta}:=\max_{|\gamma|\leqslant
N}A_{\gamma,\beta}\binom N{|\gamma|}$. We get
\begin{align*}
\lVert \widehat f\rVert_{(N,\beta)}&\leqslant B_{N,\beta}\sum_{|\gamma|\leqslant N}\:
\sum_{\alpha\leq \gamma} \lVert f\rVert_{(n+1+\beta,\alpha)}\\\
&\leqslant B_{N,\beta}\sum_{|\gamma '|\leqslant N} D(\gamma')\lVert f\rVert_{(n+1+\beta,\gamma')},
\end{align*}
where $D(\gamma')$ denote the number of times on which $\gamma'$ is
obtained in the double sum. Finally, we get
$$\lVert \widehat f\rVert_{(N,\beta)}\leqslant C_{N,\beta}\sum_{|\gamma |\leqslant N} \lVert f\rVert_{(n+1+\beta,\gamma)}$$
putting $\displaystyle C_{N,\beta}:=B_{N,\beta}\max_{|\gamma'|\leqslant N}D(\gamma')$.