All Questions
95
questions
-1
votes
2
answers
10k
views
Is a uniformly continuous functions bounded? [duplicate]
Let f be uniformly continuous on (a,b). How do you prove that it is bounded on (a,b)?
user81883
6
votes
5
answers
17k
views
If $f(x)<g(x)$ prove that $\lim f(x)<\lim g(x)$
I have this question:
Let $f(x)→A$ and $g(x)→B$ as $x→x_0$. Prove that if $f(x) < g(x)$ for all $x∈(x_0−η, x_0+η)$ (for some $η > 0$) then $A\leq B$. In this case is it always true that $A &...
- 403
13
votes
1
answer
4k
views
Lim Sup/Inf for real valued functions
To understand the notion of, say, limit superior for a sequence, is not difficult. Simply consider the set of all upper buonds for the set of all limit points of the sequence, and then simply pick the ...
- 760
28
votes
3
answers
2k
views
What kind of "mathematical object" are limits?
When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly ...
- 1,191
2
votes
1
answer
544
views
limit of Lebesgue integrable functions
Given two sequences of integrable functions $\{f_{n}\}, \{g_{n}\}$ with limits $f$ and $g$ both also integrable. Does this always hold
$$\lim_{n}(f_{n}-g_{n})=\lim_{n}f_{n}-\lim_{n}g_{n}=f-g$$
I ...
- 747