Proof of a normed space

1. cabin5

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1. The problem statement, all variables and given/known data
The norm is defined by $$\left\|x\right\|_{p}=\left[\sum^{\infty}_{k=1}\left|\alpha^{i}\right|^{p}\right]^{1/p}$$
where $$x=(\alpha^{1},\alpha^{2},....,\alpha^{n})$$

Prove that this is a norm on $$V_{\infty}(F)$$
2. Relevant equations
All conditions satisfied for a normed space.

3. The attempt at a solution
Well, I proved the first condition which is

i)
$$\left\|x\right\|_{p}=\left\{\left|\alpha^{1}\right|^{p}+\left|\alpha^{2}\right|^{p}+.....+\left|\alpha^{n}\right|^{p}+....\right\}^{1/p}$$

This must be positive definitive, therefore $$\left\|x\right\|_{p}>0$$

On the second condition I don't know whether taking the product of this norm with a $$\beta\in F$$ since the sum is infinite. I got stuck at this point.

And also I presume, for the third condition, Minkowski inequality cannot be used anymore to prove it.