Question Number 88607 by M±th+et£s last updated on 11/Apr/20
$${chose}\:{the}\:{right}\:{option} \\ $$$$\left.\mathrm{1}\right)\:{if}\:{x},{y}\:\in{Q}^{{c}} \:{and}\:{x}<{y}\:\:{then}\:\exists{a}\in{Q} \\ $$$${such}\:{that}\:{x}<{a}<{y}\:{in}\:{case}\:{we}\:{say}\:{that} \\ $$$$\left.{a}\right){Q}^{{c}} \:{is}\:{an}\:{ordered}\:{field} \\ $$$$\left.{b}\right){Q}\:{dense}\:{inQ}^{{c}} \\ $$$$\left.{c}\right){not}\:{ordered}\:{field} \\ $$$$ \\ $$$$\left.\mathrm{2}\right){let}\:{Q}\neq{A}\:\in\:{Q},{A}\:{not}\:{necessary}\:{has} \\ $$$${least}\:{upper}\:{bound}\:{and}\:{greatest}\:{lower} \\ $$$${bound}\:{that}\:{mean}\:\left({Q},+,.,\leqslant\right){is}……. \\ $$$$\left.{a}\right){not}\:{complete} \\ $$$$\left.{b}\right){complete} \\ $$$$\left.{c}\right){dense}\:{in}\:{R} \\ $$$$ \\ $$$$\left.\mathrm{3}\right){the}\:{sequence}\:{a}_{{n}} =\frac{{n}}{{n}+\mathrm{1}}\:{is}\:{convergent} \\ $$$${to}…… \\ $$$$\left.{a}\right)\mathrm{0} \\ $$$$\left.{b}\right)\mathrm{1} \\ $$$$\left.\mathrm{3}\right)\infty \\ $$$$ \\ $$$$\left.\mathrm{4}\right)\:{let}\:{f}:{A}\rightarrow{B}\:{be}\:{a}\:{real}\:{valued}\:{function} \\ $$$${and}\:{Q}\neq{S}\subseteq{A}\:{such}\:{that}\:{f}\:{is}\:{not}\:{continuous} \\ $$$${on}\:{S} \\ $$$${then}\:{f}\:{is} \\ $$$$\left.{a}\right){contiuous}\:{on}\:{A} \\ $$$$\left.{b}\right)\:{continuous}\:{at}\:{any}\:{points}\:{x}_{\mathrm{0}} \:{in}\:\:{A} \\ $$$$\left.{c}\right)\:{continuous}\:{on}\:\:{A}/{S} \\ $$$$ \\ $$$$\left.\mathrm{4}\right){every}\:{set}\:{of}\:{natural}\:{numbers}\:{has} \\ $$$${a}\:{least}\:{element}\:{so}\:{the}\:{order}\:{set}\:{of} \\ $$$${natural}\:{number}\:{is}…… \\ $$$$\left.{a}\right){bounded}\:{from}\:{above} \\ $$$$\left.{b}\right){bounded}\:{from}\:{below} \\ $$$$\left.{c}\right){well}\:{ordered} \\ $$