- #1
mathmonkey
- 34
- 0
Homework Statement
Hi I've come across the term lim inf ##f_n## in my text but am not sure what it means.
##\lim \inf f_n = \sup _n \inf _{k \geq n} f_k##
In fact, I am not sure what is supposed to be the output of lim inf f? That is, is it supposed to return a real-valued number, or a function itself?
Generally for a real-valued function ##\inf f## refers to ##\inf_x f(x)##. That is, it returns the largest real-valued number smaller than f(x) for all x. If that's the case, then it should follow that ##\lim \inf f_n## also returns a real-valued number? But I've always thought the implication of lim inf and lim sup is that if ##f_n## converges uniformly to ##f## then
##\lim \inf f_n = \lim \sup f_n = \lim f_n = f##
but that doesn't seem to hold if i use this definition? Any help or clarification would be greatly appreciated. Thanks!