Let $E=\{0,1\}^\mathbb{N}$, and $d: E\to \mathbb{R}$, defined by $d(x,x)=0$ and $$d(x,y)= 2^{-\min \left\{k\in \mathbb{N}\mid x_k \neq y_k\right\}}$$. For all $x=(x_k)_k,y=(y_k)_k \in E$, prove that $d$ is a metric.
I'm having problems with $d(x,y)\leq d(x,z)+d(z,y)$, Hints are preferred.
For any $k$, $$ x_k\neq y_k\quad \Rightarrow\quad x_k\neq z_k \quad \vee\quad y_k\neq z_k. $$ Then $$ \min\{k:x_k\neq y_k\}\geq \min\{k:x_k\neq z_k\} \quad \vee\quad \min\{k:x_k\neq y_k\}\geq \min\{k:y_k\neq z_k\} $$ hence $$ d(x,y)\leq d(x,z) \quad \vee\quad d(x,y)\leq d(y,z) $$
