@PeterTamaroff you have $\pm \sum x_i y_i \leq \left| \sum x_i y_i \right| \leq \sum |x_i||y_i| \leq \left( \sum |x_i|^p\right)^{1/p} \cdots$, so the result stating the last inequality states more than just $\sum x_i y _i \leq \left( \sum |x_i|^p\right)^{1/p} \cdots$: stronger inequality = sharper result.
suppose $A$ is a set of 16 distinct natural numbers. such that if $p\in A $ then $1\leq p\leq100$
prove there exist numbers $a,b,c$ and $d \in A$ for a, b, c and d distinct
such that a+b=c+d
@Khromonkey: Think about this. If the condition that elements of $A$ are less than $100$ is removed, can you find sixteen numbers so that FOR ALL Elements of $A$, $a+b \neq c+d$
@Khromonkey yes. I do. We have only proven that there exist two distinct sets $\{a,b\}$ and $\{c,d\}$ such that $|a-b|=|c-d|$. This means that without loss of generality $a\neq c$.
@Khromonkey I have another idea. We should try to show that there is an arithmetic progression of length $4$ in the set $A$. If that is true. It would be an easy proof then. What about MO?
@t.b. If I know that the kernel if isomorphic to $\Bbb{Z}^k$ for some $k$, then $\Bbb{R}^n/\Bbb{Z}^k \cong \Bbb{R}^k/\Bbb{Z}^k \oplus \Bbb{R}^{n-k}$ finishing the problem yes?
@t.b. One day, my first abstract algebra teacher got tired of saying the entire word "homomorphism" and said "These maps are clearly Homo and these are..." and he realized what he had said. Needless to say, the rest of the lecture was incredibly awkward.
Como estamos seguros que $$\sum_{i=2}^{n+1} \frac{\lambda_i}{1-\lambda_1} x_i$$ esta en el dominio de $\varphi$? (Para poder seguir a la siguente desigualdad) La desigualdad sigue por convexidad, pero esa suma tiene que estar en el dominio de $\varphi$)
En realidad no entiendo como aplican la hipotesis inductiva.
@leo La hipotesis es simplemente que para $n$ numeros la hipotesis es cierta. Como la suma de $i=2$ a $n+1$ es de $n$ numeros y la suma es $1$, la induccion demustra el teorema, si?
con lo del dominio, dado $D\subseteq \Bbb R$ convexo, si $x_1,\ldots,x_n\in D$ y $\sum_{i=1}^n \lambda_i$ con $\lambda_i\in [0,1]$ entonces $\sum_{i=1}^{n}\lambda_ix_i\in D$
suppose $A$ is a set of 16 distinct natural numbers. such that if $p\in A $ then $1\leq p\leq100$
prove there exist numbers $a,b,c$ and $d \in A$ for a, b, c and d distinct
such that a+b=c+d
