# real analysis: prove that a set A is open with respect to the metric standard dp if and only if it is open with dq

Consider two metrics $$d_ {p} = ( sum limits_ {k = 1} ^ n | x_ {k} -y_ {k} | ^ p) ^ {1 / p}$$ Y $$d_ {q} = ( sum limits_ {k = 1} ^ n | x_ {k} -y_ {k} | ^ q) ^ {1 / q}$$

Prove that a non-empty subset $$A subset mathbb R ^ n$$ is open with respect to $$d_ {p}$$ If it is open with respect to $$d_ {p}$$

Tried

So $$A$$ It's open both in those metrics if $$d_ {p}$$ Y $$d_ {q}$$ they are equivalent

Then I have to show that $$alpha d_p (x, y) leq d_q (x, y) leq beta d_p (x, y)$$ for $$alpha,$$ positive.

How do I begin to demonstrate this?