Prove the following metrics are equivalent:
$$d_1(f,g) = \sup_{N \geq{0} } \frac{1}{2^N} \frac{\| f-g \|_N}{1 + \| f-g \|_N}$$ and $$d_2(f,g) = \sum_{N = 0}^{\infty} \frac{1}{2^N} \frac{\| f-g \|_N}{1 + \| f-g \|_N}$$ where $$\| \phi \|_N = \max _ {x \in \Omega} \{ |D^\alpha \phi(x) | : |\alpha| \leq N \}$$
with $$\alpha$$ is a multi-index
Any hint? What I tried is to see if they are in strong equivalence (because I thought it'd be easier) but I didn't get nothing interesting. Useful information: $f$ and $g$ are smooth functions with compact support. I'm working over the space of test functions.
Thanks in advance.