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Results tagged with real-analysis
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user 1046392
For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.
3
votes
continuity (by definition) at a point on piece-wise functions
A sequence of real numbers $(a_n)$ satisfies
$$ (a_n)\to 0 \iff (|a_n|)\to 0 $$
So to prove that $$(f(x_n)) \to 0$$
It is sufficient to prove that
$$(|f(x_n)|)\to 0 $$
But for any $x$, $$|f(x)| = 2x^2 …
- 3,044
1
vote
Accepted
Real Analysis: equivalence of a limit related statement
To point out this equivalence is not a good approach to solving this limit, especially for students still coming to terms with the concept of limits, for the following reason:
Although the two stateme …
- 3,044
1
vote
why $f_2^{-1} \circ f_1(x) = \sqrt [3]x ?$
This is a fairly simple question about functional composition, but I understand that when one learns more advanced topics you might get a bit fuzzy on some of the basics, so:
since $f_2(x) = x^3, f_2^ …
- 3,044
1
vote
Accepted
Why are Diagonals Special: Theorem 2.12 in Rudin PMA
Perhaps the double indexing in the matrix construction is what is confusing you, if this is the case I recommend you think of this in a different way.
You know (I presume) that $\mathbb{N}^2$ is count …
- 3,044
3
votes
Accepted
Proof that $\sup \frac{1}{A} = \frac{1}{\inf A}$
Suppose $\frac{1}{\alpha}$ is not the least upper bound, then there is some other upper bound, call it $\frac{1}{a'} $ for which
$$0<\frac{1}{a'}<\frac{1}{\alpha} $$
Then since $\frac{1}{a'}$ is an up …
- 3,044
3
votes
Is the sequence of functions uniformly convergent in $[0,1)$?
Your initial guess is incorrect, these functions are essentially just step functions, where the step size gets smaller and smaller as the number of steps gets smaller and smaller, this is evident from …
- 3,044
1
vote
1
answer
36
views
The building blocks of $\mathbb{R}$
In an introductory class to analysis, a theorem we prove is the characterization of intervals (I'm sure there is more than one by this name, but its not important for this question) but we don't prove …
- 3,044
2
votes
Accepted
Is the 'greatest negative number' indeterminate, not defined or doesn't exist?
The notion of a "greatest element" does not only make sense for finite sets, without defining posets, lets just focus on the real numbers:
Given a set $A\subseteq\mathbb{R}$, a real number $x$ is sai …
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1
vote
Simple functional equation: $f(x)=f(y) \implies f(ax)=f(ay)$
Define a cone of $\mathbb{R}^n$ as a set that can be written as the union the positive spans of vectors (we will call them positive rays) in $\mathbb{R}^n$. By positive span of a vector $\mathbf{x}$ I …
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2
votes
0
answers
38
views
Defining the real numbers up to isomorphism
One of the standard definitions for the real numbers, $\mathbb{R}$, uses the fact that any set $S$ which is a complete, ordered archimedean field, is field- and order-isomorphic to the real numbers. B …
- 3,044
2
votes
Accepted
Proving $f(x) = \frac{1}{x}$ is not uniformly continuous on (0,1]
Unfortunately, this proof is not correct. You mentioned that it didn't look convincing, and you had good intuition! Now lets try to unpack why this didn't feel convincing.
If you think of a proof like …
- 3,044
4
votes
1
answer
130
views
An alternative definition of the limit of a function
In a course on real analysis one usually comes across the definition of the limit of a function:
Given a function $f:A\to \mathbb{R}$ where $A\subseteq\mathbb{R}$, then if $c\in A$ is an limit point o …
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0
votes
1
answer
62
views
Is there a proof that continuous linear operators are bounded that uses this line of reasoning?
I am doing an introductory functional analysis course alongside a real analysis course, and the following concept showed up. I am looking for a proof that combines what I have learned from these two c …
- 3,044
2
votes
Can $⟨a_n⟩$ be oscillatory if $\lim_{n\rightarrow\infty}a_{n+1}/a_n = 1$?
Yes this is possible, depending on what you mean by oscillatory
If you maintain that the values "oscillate" around the value 0, then this isn't possible, since we would then have
$$ sgn(T_{n+1}) = -sg …
- 3,044
1
vote
1
answer
84
views
Confusion about why we need compactness
In my real analysis course, we need to prove that if $K\subseteq\mathbb{R}$ is compact, and $f:K\to\mathbb{R}$ is continous and injective, then $f^{-1}$ is continuous. The standard proof for this uses …
- 3,044