We know that every Cauchy sequence is bounded. and the converse may not be true. but if we define a sequence $x_n=n$ with metric $d(m,n)$=$\lvert \dfrac{1}{m}-\dfrac{1}{n}\rvert$. then this is a Cauchy sequence but our sequence was unbounded.
My question is that does necessary condition of boundedness gets affected if we change the metric condition on the space.
Your sequence is bounded, since $(\forall n\in\mathbb{N}):d(x_n,1)\leqslant1$.
On the other hand, yes, a set being bounded depends upon the metric, as your example shows.