inequality in a banach space

Question

Let $(X,\left\| \cdot \right\|)$ be a Banach space. For each $i=1,\cdots,n$, let $a_i\in X$ and $\alpha_i\in\mathbb{R}$. Suppose that $0\leq\alpha_i\leq M$ for all $i=1,\cdots,n$.

Question: Is it true that $$\left\|\sum_{i=1}^n a_i\alpha_i\right\|\leq M\left\|\sum_{i=1}^n a_i\right\|?$$ I dont have any idea if the above inequality is true. In case it is true, can you please give me some hints on how to prove it? Many thanks in advance.

Answer 1

This is false. Take $X =\Bbb{R}$, $n=2$, $M=1$, $a_1=1$, $a_2=-1$ and $\alpha_1=1/2$, $\alpha_2=1$.

Then the right hand side is $0$, but the left hand side is

$$ \Vert \sum \alpha_i a_i\Vert =| 1/2 - 1| =1/2. $$