My book (Principles of Topology by Fred Croom) states the following lemma: "Let $d_1$ and $d_2$ be two metrics for the set $X$ and suppose that there is a positive number $c$ such that $d_1(x,y) \le cd_2(x,y)$ for all $x,y\in X$. Then the identity function $i: (X,d_2) \to (X,d_1)$ is continuous."
My question is simple: must $c$ be a constant independent of $x$ and $y$, or is it a not-necessarily-constant positive number? Experience suggests the former, but the lack of the word 'constant' and two possible interpretations of the wording make me question it. I worked out a proof that every metric space is homeomorphic to a bounded metric space in which it works to set $c=d(x,y)$, and this is the reason that I want to get this straight.