Problem: Let $(X,d)$ be a metric space and $B(x,r)$ denotes the open ball in $X$, that is, \[ B(x,r) = \left\{ y\in X: d(x,y) \lt r \right\} . \] Let $y,z\in X$ and $x \in B(y,\alpha ) \cap B(z,\beta )$, where $\alpha ,\beta $ are positive real numbers. Then show that there exists $\epsilon >0$ such that $B(x,\epsilon ) \subseteq B(y,\alpha ) \cap B(z,\beta )$.