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?