Problem : Metrics and Induced Topologies

[AKG]

The Euclidean metric, d, is defined by:

$$d(x, y) = \left [\sum _{i = 1} ^n |x_i - y_i|^2\right ]^{1/2}$$

Define metrics d_p for each p in {1, 2, 3, ...} as follows:

$$d_p(x,y) = \left [\sum _{i = 1} ^n |x_i - y_i|^p\right ]^{1/p}$$

Prove that each d_p induces the same topology as the Euclidean metric.

To do this, I want to show that for every $\epsilon > 0$ and for every $x \in \mathbb{R}^n$, there is are $\delta _1,\, \delta _2 > 0$ such that for every $y \in \mathbb{R}^n$:

$$\left [\sum _{i = 1} ^n |x_i - y_i|^2\right ]^{1/2} < \delta _1 \Rightarrow \left [\sum _{i = 1} ^n |x_i - y_i|^p\right ]^{1/p} < \epsilon$$

and

$$\left [\sum _{i = 1} ^n |x_i - y_i|^p\right ]^{1/p} < \delta _2 \Rightarrow \left [\sum _{i = 1} ^n |x_i - y_i|^2\right ]^{1/2} < \epsilon$$

Is this the right way to prove it? Where do I go from here? Induction on n, or p? Or maybe both? Or is there a way to do it without induction? Help would be very much appreciated!

 

[AKG]

Actually, the problem I really have to solve is to show that, assuming each d_p is a metric, they all induce the usual topology on Rⁿ, and I figured the best way to do this was to show that they induced the same topology as the Euclidean metric since these "metrics" (they might not all be metrics, but the problem says to assume they are) look a lot like the Euclidean metric.

 

[Hurkyl]

Well, when you don't understand something, draw a picture.

A circle (or an n-sphere, in general) is a characteristic of the Euclidean metric, right? What about these other metrics?