- #1
mathboy20
- 30
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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]
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
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
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