Theorem 1.3.5 p19 in "the analysis of linear partial differential operators I" by Hormander states that for any positive sequence $a_{0}\geq a_{1}\geq \cdots $ such that $a=\sum_{k=0}^{+\infty }a_{k}<+\infty $, if $% u_{k}=H_{a_{0}}\ast \cdots \ast H_{a_{k}}$ (where $H_{c}(x)=c^{-1}$ when $% 0\leq x\leq c$ and $H_{c}(x)=0$ otherwise), then $u_{k}\in $ $% C_{0}^{k-1}\left(\mathbb{R} \right) $ has support in $\left[ 0,a\right] $ and converges as $k\rightarrow +\infty $ to a function $u\in C_{0}^{\infty }\left( \mathbb{R}\right) $ with support in $\left[ 0,a\right] $ such that $\int udx=1$ and $% \left\vert u^{(k)}(x)\right\vert \leq 2^{-1}\int \left\vert u^{(k+1)}(x)\right\vert dx\leq 2^{k}(a_{0}...a_{k})^{-1},k=0,1,\ldots $
In the proof of the theorem, the author shows clearly that $\left\vert u^{(k)}(x)\right\vert \leq 2^{k}(a_{0}...a_{k})^{-1}$ and $2^{-1}\int \left\vert u^{(k+1)}(x)\right\vert dx\leq 2^{k}(a_{0}...a_{k})^{-1}$, but, if I am not wrong, he doesn't give any hint for the inequality $\left\vert u^{(k)}(x)\right\vert \leq 2^{-1}\int \left\vert u^{(k+1)}(x)\right\vert dx$. I have tried to state it myself but all I have found is the inequality : $ \int \left\vert u^{\left( k\right)}\left(x\right)-u^{\left( k\right)}\left(x-a_{k}\right)\right\vert dx \leq 2^{-1}\int \left\vert u^{(k+1)}(x)\right\vert dx$ and nothing further. Please help me clarify this point, as I am totally stuck. Thanks in advance.