How can I show that $ \sum_{k=0}^{m}\binom{n}{k} \leq (n+1)^m$ for $m \leq n $?
I have to prove by induction that $$\sum_{r=0}^{n}\begin{pmatrix}n \\ r \end{pmatrix}=2^k$$, using the identity $\begin{pmatrix}n \\ r-1 \end{pmatrix}+\begin{pmatrix}n \\ r \end{pmatrix}=\begin{pmatrix}n+1 \\ r \end{pmatrix}$. I'm fine with everything except the inductive step, where I only managed to get as far as $$\sum_{r=0}^{k+1}\begin{pmatrix}n \\ r \end{pmatrix}= \sum_{r=0}^{k}\begin{pmatrix}n \\ r \end{pmatrix}+\begin{pmatrix}n \\ k+1 \end{pmatrix}$$ I tried to then use the identity and substitute $\begin{pmatrix}n \\ r \end{pmatrix}=\begin{pmatrix}n+1 \\ r \end{pmatrix}-\begin{pmatrix}n \\ r-1 \end{pmatrix}$ but all this did was just take me around in a circle. I also couldn't substitute $2^k$ for $\sum_{r=0}^{k}\begin{pmatrix}n \\ r \end{pmatrix}$ since this didn't properly simplify to give $2^{k+1}$. However, I think that I have to obtain the expression $\sum_{r=0}^{k}\begin{pmatrix}n \\ r \end{pmatrix}$ since it simplifies to $2^{k+1}$. But I've not been able to obtain this in anyway so I'm wondering if anyone can guide in what I need to or what direction I need to take to complete the inductive step.
postfix fails to send mail with "fatal: unknown service: smtp/tcp" I wanted to show that $X' / U^\perp \cong U'$, for $U$ being a closed subspace of the Banach space $X$.  Therefore I looked at $l: X' / U^\perp \cong U' , x' + U^\perp=[x']  \mapsto x'|_U$.  It is clear that this map is onto, as we just take a $u' \in U'$. Then I can use Hahn-Banach to get an appropriate $x'$ such that $x'|_U = u'$ and in particular $||x'||=||u'||$.  Now I need to show that is isometric: since we always have $\|[x']\|\le \|x'\|$ and by Hahn-Banach we have $\|x'\|=\|x'_{|U}\|$, this shows $\|[x′]\|\le \|x'_{|D}\|$ and it should be always true that $\forall y' \in U^\perp: \|x'-y'\| \ge \|x'_{|D}\|$, hence $\|[x']\|\ge \|x'_{|D}\|$. Is this reasoning correct? The isometric property follows. 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