Triangle inequality for R^n

How would I show the triangle inequality for $\displaystyle \mathbb{R}^{n}$ ?

The only thing I can think of is induction and thinking about it, it doesnt make sense that the triangle inequality in $\displaystyle \mathbb{R}^n$ depends on the triangle inequality in $\displaystyle \mathbb{R}^2$

Re: Triangle inequality for R^n

Quote:

Originally Posted by

**Jame** How would I show the triangle inequality for $\displaystyle \mathbb{R}^{n}$ ?

The only thing I can think of is induction and thinking about it, it doesnt make sense that the triangle inequality in $\displaystyle \mathbb{R}^n$ depends on the triangle inequality in $\displaystyle \mathbb{R}^2$

Why would you say that?

Is it not the case that $\displaystyle \mathbb{R}^3=\mathbb{R}^2\times\mathbb{R}~?$

Thus $\displaystyle \mathbb{R}^{n+1}=\mathbb{R}^n\times\mathbb{R}~.$