Mathematical and real analysis

Question

Let $d(A,B)=\inf\{d(x,y)\mid y\in B, x\in A\}$ that $A$ and $B$ are subset of metric space $X$. If $d(A,B)\leq 1$, then $d(x,y)\leq 1$ for every $x\in A, y\in B$?

Answer 1

What you're asking is "If the infimum of some set of real numbers is less than $1$, then are all elements of that set less than $1$?" The answer to that is no in general, although for some specific $A$ and $B$ it is true.