sup of a sequence of functions
let {f_n} n=1 to infinity, be a sequence of functions
what does do the following mean
(1) sup_n f_n (x)
(2) lim sup_n f_n(x)
i know the sup of a function (it is the number in the domain which
gives the highest value in the range)
but the sup of the sequence has me quite confused. I would really
appreciate a little explanation of these 2.
.
Relevant Pages
- Re: Idiosyncratic (or idiotic?) notation for lim inf, etc.
... >All the results concern a sequence of integrable functions. ... >use `Sup s' and `Inf s' to denote the upper and lower sequences ... >lim always exist, ...
(sci.math) - Idiosyncratic (or idiotic?) notation for lim inf, etc.
... All the results concern a sequence of integrable functions. ... Then Sup s is a decreasing sequence, and Inf s an increasing ... lim always exist, ...
(sci.math) - Re: Prove Sup a_n=Inf b_n for (a_n,b_n) sequence of nested intervals
... Let be a sequence of nested open intervals. ... Show that Sup a_n=Inf b_n ... approaches zero as becomes a smaller and smaller interval as a_n and b_n get closer and closer for each successive nested interval. ...
(sci.math) - Re: Prove Sup a_n=Inf b_n for (a_n,b_n) sequence of nested intervals
... Let be a sequence of nested open intervals. ... Show that Sup a_n=Inf b_n ... approaches zero as becomes a smaller and smaller interval as a_n and b_n get closer and closer for each successive nested interval. ...
(sci.math) - Re: "limit"
... The limit does exist and equal to infinity. ... when dealing with limits, we think of 'extended real number system', ... " lim f= -oo ... relate the limit of a function to a limit of a sequence ("as n, ...
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