Cauchy sequence with respect to Hausdorff metric

Question

We know that if $(X,d)$ is a complete metric space, then $(CB(X),H)$ is complete too, where $CB(X)$ is the collection of non-empty closed bounded subset of $X$ and $H$ is the Hausdorff metric induced by $d$. Let $T:X\to CB(X)$ be a mappings. Does all Cauchy sequence $\{x_n\}$ in $(X,d)$ implies that $\{Tx_n\}$ is a Cauchy sequence in $(CB(X),H)$? and the converse is it true, that is, does all Cauchy sequence $\{Tx_n\}$ in $(CB(X),H)$ implies that $\{x_n\}$ is a Cauchy sequence in $(X,d)$?

Answer 1

If $T:(X,d)\to (CB(X),H)$ is a natural mapping $x\mapsto \{x\}$ then $T$ is an isometry.