# Problem : Metrics and Induced Topologies

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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 dp 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 dp 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!

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