Prove that every metric space is homeomorphic to a bounded metric space. The sample solution in the book is: $d'(x,y)=\mbox{min}\{1,d(x,y)\}$.
However, I came up with a different solution and wonder if it's correct: $d'(x,y)=\frac{d(x,y)}{1+d(x,y)}$. Clearly it's $\leq 1$. As the homeomorphism I choose the identity. It's obviously continuous in both directions.