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Suppose $A$ is a normed vector space and $a_i \in A$ and we have the following sequence $ \Vert a_1\Vert \leq \Vert a_2 \Vert \leq \cdots \leq \Vert a_n \Vert $

for any $1\leq i. How can I show the following is not necessarily true $\Vert a_{i+1} -a_i \Vert \leq \Vert a_{i+2} - a_i \Vert ?$ It's easy to see this is not true for $\mathbb{R}$ any other ways of seeing this.

Thanks.

3 Answers 3

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it easy to see this is not true for R any other ways of seeing this

If you already know it is not true for $\mathbb{R}$, you can take any counterexample in $\mathbb{R}$ and multiply by a fixed nonzero vector in $A$ (assuming $A$ has a nonzero vector, of course). For example, multiplying $1,-1$, and $2$ by $x\neq 0$ gives $\|x\|=\|-x\|\lt\|2x\|$ but $\|-x-x\|\gt\|2x-x\|$.

But perhaps by "other ways" you meant that you had already considered this method and wanted a different type of example. In that case, Rasmus's nice example is definitely a better answer.

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    @Rasmus: Thanks. Your example is nice, though, as it shows that you don't have use vectors going in opposite directions to get examples. I don't know, maybe $1,-1$, and $1$ is more dramatic.2011-01-29
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Let $A$ be $\mathbb R^2$ with the Euclidean norm and let $n=3$.

Let $a_1=(0,1)$, $a_2=(1,0)$, and $a_3=(0,2)$.

Then $a_1$ and $a_2$ have norm $1$ and $a_3$ has norm $2$. That is, your condition on the norms is fulfilled.

However, $\Vert a_2-a_1\Vert=\sqrt{2}>1=\Vert a_3-a_1\Vert$.

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    That's e$x$actl$y$ an illustration of my $c$omment, thanks!2011-01-29
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It mostly relies on the triangle inequality.

If you have $||a_1|| \leq \cdots \leq ||a_n||$ then by substracting by $||a_i||$ for a given $i$ gives : $0 \leq ||a_{i+1}|| - ||a_i|| \leq ||a_{i+2}|| - ||a_i||$

But the triangle inequality only tells you $||a_{i+1}|| - ||a_i|| \leq ||a_{i+1} - a_i||$ and $||a_{i+2}|| - ||a_i|| \leq ||a_{i+2} - a_i||$ respectively. I don't see any way to go any further.

You can visualize this a bit: imagine $a_i$ and $a_{i+1}$ have close norms but "point to" very different directions, and that $a_i$ and $a_{i+2}$ have very different norms but sensibly the same direction. Then in that case you can see it wouldn't work.