Let $x,y,zin M:={ 0,1}^{mathbb{N}}$ and define $mu(x,y)=min{ nin mathbb{N}mid x_{n}neq y_{n}}$.
I want to show that $mu(x,z)geq min left { mu(x,y),mu(y,z) right }$.
I have tested that it is true if I, for example, let $n=3$. Now I want to prove it generally. If I assume that $x=y=z$ then it is clearly satisfied. Assuming $xneq yneq z$ seems challenging to me. Since $min left { cdot,cdot right }$ has only one value then I can assume without loss of generality that $min left { mu(x,y),mu(y,z) right }=mu(x,y)$. I am thinking about proving it by contradiction that $mu(x,z)<mu(x,y)$ for all $x,y,zin M$ but I am unsure about it.
