(Solved): 1. (a) State order completeness property of real numbers.Prove that the set of rational numbers is ...

1. (a) State order completeness property of real numbers.Prove that the set of rational numbers is not order complete. (b) Prove that the intersection of arbitrary family of open set need not be open. 2. (a) Prove that every infinite bounded set of $\mathrm{R}$ has at least one limit point. (b) Define neighbourhood of a point. Show that, if $\mathrm{M}$ and $\mathrm{N}$ are neighbourhoods of a point $\mathrm{p},M?N$ is also neighbourhood of $\mathrm{p}$. 3. (a) State and prove Cauchy's first Theorem on limits. (b) Prove that the sequence $?a_n?$ defined by $a_1=\sqrt{7},a_{n+1}=\sqrt{7+a_n}$ converge to the positive root of the equation $x^2?x?7=0$.