Define a1=1 and an+1=1/2(an+3/an) for n≥1.

By AM-GM, $a_{n+1}=\frac12\left(a_n+\frac3{a_n}\right)\ge\sqrt{3}$ for every $n\ge1$, with equality only if $a_n=\sqrt3$, which never happens because all terms are rational. Hence $a_n>\sqrt3$ for $n\ge2$, so (D) is false. For $n\ge2$ we have $a_n^2>3$, so $a_{n+1}-a_n=\frac{3-a_n^2}{2a_n}<0$: the sequence decreases from $n=2$ onwards and, being bounded below by $\sqrt3$, converges to a limit $\ell$ with $\ell=\frac12(\ell+3/\ell)$, giving $\ell=\sqrt3$; so (A) is true. However $a_1=1<a_2=2$, so the whole sequence is not monotonically decreasing and (B) is false. Computing exactly, $a_2=2$, $a_3=\frac74$ and $a_4=\frac12\left(\frac74+\frac{12}{7}\right)=\frac{97}{56}$, not $\frac{26}{15}$, so (C) is false.