In solving exercise 8.6.5 in Brezis' Functional Analysis, I have to prove an auxiliary result, i.e.,
Let $I$ be an open interval of $\mathbb R$ and $m \ge 1$ an integer and $p \in [1, \infty]$. The injection $W^{m, p}(I) \subset C^{m-1} (\bar{I})$ is continuous.
There are possibly subtle mistakes that I could not recognize in below attempt. Could you have a check on it?
We proceed by induction on $m$. The base case $m=1$ has been proved in Theorem 8.8 (in the same book). Assume that the injection $W^{k, p} (I) \subset C^{k-1} (\bar{I})$ is continuous for all $1 \le k \le m$. We will prove that the claim holds for $k=m+1$.
Let $u \in W^{m+1, p} (I)$. Then $u \in W^{m, p} (I)$ and $D^m u \in W^{1, p} (I)$. By induction hypothesis, there are constants $c_1, c_2 >0$ such that $|u|_{C^{m-1} (\bar I)} \le c_1 |u|_{W^{m, p}}$ and $|D^m u|_{C^0 (\bar I)} \le c_2 |D^m u|_{W^{1, p}}$. We have $$ \begin{align} |u|_{C^m (\bar I)} &= |u|_{C^{m-1} (\bar I)} + |D^m u|_{C^0 (\bar I)} \\ &\le c_1 |u|_{W^{m, p}} + c_2 |D^m u|_{W^{1, p}} \\ &\le c_1 |u|_{W^{m+1, p}} + c_2 | u|_{W^{m+1, p}} \\ &= (c_1 + c_2) |u|_{W^{m+1, p}}. \end{align} $$
The claim then follows.
