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Results tagged with real-analysis
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user 700617
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.
1
vote
1
answer
48
views
Proof that $f′(0)=0$ when $x=0$ is a local extremum(taylor method)
I am working with the proof that $f′(0)=0$ when $x= 0$ is a local extremum, for the taylor method (I understand the Fermat's theorem)
I was given an answer this question on here
Assume $f$ has a lo …
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0
votes
0
answers
45
views
Question on Negating "for all"
I understand that for negation $\forall$ converts into $\exists$
like: $\forall x \ , \forall y$, $ \ y^2 + x^2 \ge0$
and the negation will be: $\exists x \ $and $ \exists y$, $ \ y^2 + x^2 < 0$
I und …
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0
votes
1
answer
55
views
(rudin theorem 3.17) Struggling understanding a part of the proof
I need help understanding a part of the proof of Theorem 3.17 in Rudin Real Analysis
The full theorem can be found here:
Is Rudin being redundant in this proof?
The problem is on this part
If $s^* = …
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3
votes
1
answer
67
views
Help with understanding Cauchy sequences?
I am trying to understand Cauchy sequences a little better and would really appreciated any insight/advice you can offer:
Defition: A sequence $\{x_n\}$ in a metric space $(X,d)$ is called a Cauchy se …
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2
votes
1
answer
226
views
prove that $\operatorname{Int}(A \cap B)= \operatorname{Int}(A) \cap \operatorname{Int}(B)$
In the book that I am using they left the proof to the theorem "as a exercise" so I would like to give it a go. Please let me know if I am incorrect.
Let $A$ and $B$ be subsets of a metric space $X$.
…
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2
votes
2
answers
93
views
Why is there a change in inequality from ">" to "$\geq$" in this example?
I copied the example out but I am not interested in the example per say, except for the change in inequality that takes place. I underlined it in red.
The Example
My Question
What I understand: (for …
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1
vote
2
answers
91
views
Help with a simple theorem proof involving supremum
Here is the theorem(just how they gave it, no details left out):
(a) $\alpha = sup(S) \iff (i)\ \alpha \ge x$ $\forall x \in S$; and
$\qquad\qquad\qquad\qquad\ \ \ \ \ (ii)\ \forall a < \alpha , \ex …
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0
votes
1
answer
118
views
Question on: it holds "trivially".
I am not really interested in the proof per say, but only the red underlined part. I have some idea how to do it but I just want to make sure. It feels like it should be obvious to me by now but still …
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0
votes
1
answer
63
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Show that $(X,d)$ is complete if and only if $(X,\overline{d})$ is complete (Part 2)
I reworked my proof that I attempted here
I used subsequences to prove it. I am not 100% sure that it is worded correctly.
I would really appreciate it if you could offer any insight/advice that you c …
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1
vote
1
answer
41
views
Suppose that $|a_n|<2$ and $|a_{n+2}−a_{n+1}|\leq\frac{1}{8}|a_{n+1}^2−a_n^2|$, prove that $...
This is my second attempt. The first attempt is here
I included the estimate $|a_m−a_n|\leq |a_m−a_{m-1}| + |a_{m-1}−a_{m-2}| \dots |a_{n+1} a_n|$
I really appreciate any advice\insight you can offer
…
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0
votes
0
answers
25
views
Help with the theorem: B(V,W) is complete if W is
I having some trouble with the proof of $\mathbf{B}(V,W)$ being a Banach space if $W$ is, where $\mathbf{B}(\cdot,\cdot)$ is the space of bounded linear operators $V\to W$.
I know there are other post …
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1
vote
0
answers
16
views
$f[a_1] = G$ means $a_1 \subseteq f^{-1}[G]$ for functions on metric spaces?
The theorem/proof:
Question
The part that I don't under stand is the red underlined bit.
What I understand:
let $a_1,a_2$ be sets s.t $a_1 \oplus a_2 = A$ and let $f[a_1] = G$ and let $f[a_2] = H$.
T …
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0
votes
1
answer
55
views
Potential problem with a proof involving Riemann Integral
My Question
Is there a problem with this proof(red underlined part)? I do apologize for the use of images.
I believe that $\alpha$ is allowed to be discontinuous at $v_i$ and $u_i$ as we are only conc …
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2
votes
1
answer
42
views
Show that a subset of $X$ is open in $(X,d)$ if and only if it is open in $(X,\rho)$
I have spent many hours trying to prove this question and only managed to come up with this dodgy "proof". To be honest I have no idea where to start and 99.99% sure that my attempted proof is not goo …
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1
vote
0
answers
78
views
Suppose that $|a_n|<2$ and $|a_{n+2}−a_{n+1}|\leq\frac{1}{8}|a_{n+1}^2−a_n^2|$, prove that $...
I have been trying to answer this question and manage to come up with this(not sure if it is correct). I would like to know if I can use $\epsilon > \frac{1}{2^{n-1}}|a_2−a_1|$ in it? it seems a bit e …
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