Let F be the label of an non-empty set and let [tex](B_m)_{m \geq 1}[/tex] be elements in [itex]2^F[/itex](adsbygoogle = window.adsbygoogle || []).push({});

Then I need to prove the following:

[tex]\mathrm{lim}_{m} \ \mathrm{sup} \ B_{m} = \mathrm{lim}_{m} \ \mathrm{inf} \ \mathrm{B_m} = \cup _{m= 1} ^{\infty} B_{m}[/tex]

if [tex]B_{m} \uparrow[/tex] which implies that [tex]B_{m} \subseteq B_{m+1}[/tex] for all [tex]m \geq 1[/tex] and

[tex]\mathrm{lim}_{m} \ \mathrm{sup} \ B_{m} = \mathrm{lim}_{m} \ \mathrm{inf} \ \mathrm{B_m} = \cap _{m= 1} ^{\infty} B_{m}[/tex]

if [tex]B_{m} \downarrow[/tex] which means that [tex]B_m \supseteq B_{m+1}[/tex] for all [tex]m \geq 1[/tex]

how do I go about proving this? Do I need to show the infimum of F first?

Sincerely

mb20

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: HELP: Set theory Question

**Physics Forums | Science Articles, Homework Help, Discussion**