Showing Euclidean metric and metric in $R^{2}$ produces same topology 
Question:
  Prove that the Euclidean metric and the metric $d_\infty \left ( x,y \right ):=\max\left \{ \left | x_{1}-y_{1} \right |,\left | x_{2}-y_{2} \right | \right \}$ defines the same topology in $\mathbb{R}^{2}$.

The Euclidean metric is defined as 
$d_2:\mathbb{R}^{n} \times \mathbb{R}^{n}\rightarrow \mathbb{R}$
$\vec{X} \times \vec{Y} \mapsto d\left ( \vec{X},\vec{Y} \right ) $
I would like to request for a useful hint to kickstart my attempt.
Thanks in advance.

Hint
To prove that two metrics define the same topology, it is sufficient to prove that those are equivalent.
