P-Product Metrics on Real Vector Space are Topologically Equivalent

Theorem
Let $$A = \reals^n$$ be an $n$-dimensional real vector space.

Let $$d_1, d_2, \ldots, d_\infty$$ be the generalized Euclidean metrics.

Then all of $$d_1, d_2, \ldots, d_\infty$$ are topologcally equivalent.

Proof
Let $$r \in \mathbb{N}: r \ge 2$$.

Let $$d_r$$ be the metric defined as $$d_r \left({x, y}\right) = \left({\sum_{i=1}^n \left|{x_i - y_i}\right|^r}\right)^{\frac 1 r}$$.