If $A$ is finite, then it can be described by a sentence $\forall v_{1}(P_{A}v_{1}\rightarrow(v_{1}=a_{1}\vee\ldots\vee v_{1}=a_{n}))$ , which is true in $\mathfrak{R}$ , and, hence, in $^{*}\mathfrak{R}$ , implying $^{*}A=A$ . If $A$ is unbounded, then the sentence $\forall v_{1}\exists v_{2}(P_{A}v_{2}\wedge|v_{2}|>|v_{1}|)$ is true in $\mathfrak{R}$ , and, hence, in $^{*}\mathfrak{R}$ , implying, in particular, that for any infinite $x\in{}^{*}\mathfrak{R}$ , there is $b\in{}^{*}A$ such that $|b|>|x|$ , i.e. $b\notin\mathbb{R}$ , and $b\in{}^{*}A-A$ . Finally, if $A$ is bounded but infinite, then, according to Exercise 5, there is a point $p\in\mathbb{R}$ such that $p$ is infinitely close to some $b\in{}^{*}A$ and $p\neq b$ , implying that $b\notin\mathbb{R}$ , and $b\in{}^{*}A-A$ .