6h
comment Are there infinitely many prime numbers?
See this.
11h
comment $\int_{A}{f_{n}} \> d\mu \rightarrow \int_{A}{f} \> d\mu$ for each $A \in \mathfrak{M}$ given certain conditions
See this.
13h
comment Evaluating an improper integral with limits $_{-\infty}^\infty$
See this.
15h
comment Without Lebesgue
Are you considering Riemann integrals only?
15h
comment Boundedness of a certain function defined on a closed bounded real interval
I think this is the best way to do it; you will use compactness or one of its equivalent forms somewhere. Here's the sequential version: assume $f$ is unbounded. Take $(x_n)$ with $f(x_n)\rightarrow \infty$. Choose $x\in I$ a limit point of $(x_n)$. Consider the behaviour of $f$ on $V_{\delta_x}(x)\cap I$.
15h
comment Boundedness of a certain function defined on a closed bounded real interval
What do you mean by "avoid that..."? What do you know about closed, bounded intervals? Have you studied compactness? Sequential compactness?
15h
comment Boundedness of a certain function defined on a closed bounded real interval
For each $x$, choose a corresponding nhood $V_{\delta_x}$. The collection of all such nhoods covers $I$. The sup of the maximums associated to the $V_{\delta_x}$ may be infinite. If only you could find a finite subcollection that covers...
17h
comment How to prove this limit in $\ell_1$
If $A$ is the LHS and $B$ is the RHS, show that given $\epsilon>0$ you have $A\ge B-\epsilon$. Towards this end, note that the mass of $w$ is "mostly achieved" on its first $N$ coordinates for $N$ big, while the mass of the $z_n$ and $w_n$ are mostly achieved on the coordinates $>N$ for $n$ large.
17h
comment Does the boundaries of non-disjoint sets in Euclidean space have common element?
$(0,2)$ and $(1,3)$.
17h
comment Does the boundaries of non-disjoint sets in Euclidean space have common element?
Consider $(-1,1)\subset(-2,2)$.
1d
comment Monotone Convergence Theorem for Riemann Integrable functions
Here's one proof (not using the hint).
1d
comment Alternate series
possible duplicate of Does $\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}$ converge conditionally?
1d
comment Who invented or used very first the double lined symbols $\mathbb{R},\mathbb{Q},\mathbb{N}$ etc
The story I heard somewhere (Knuth's Texbook?) is that they are as so because forming boldface symbols on a chalkboard is difficult; hence "blackboard bold".
Jul
19
comment Intersection of countable many sets of measure $1$
What's the measure of the complement of $\bigcap\limits_{n\in\Bbb N} A_n$?
Jul
19
comment prove ${a_n} = \root n \of {n!} $ is monotonically increasing to $\infty$
See this for 2..
Jul
19
comment Series in a space which is not complete
If $(x_n)$ is Cauchy, it has a subsequence $(y_n)$ with $\Vert y_{n+1}-y_n\Vert<1/2^n$. Consider the series $y_1+(y_2-y_1)+(y_3-y_2)+\cdots$. It converges if and only if $(y_n)$ is convergent.
Jul
18
comment Find all the subsets bounded and nonempty such that sup(A)‚ȧinf(A)
What can you say if $A$ has two distinct points?
Jul
18
revised Solve x in logarithm equation
edited tags
Jul
18
comment Flaw in the proof that a set is countable
What about infinite sequences (that aren't eventually constant)?
Jul
18
comment Problem in functional analysis.
You should consider the operator from $X^*$ to $\ell_1$ that maps $f\in X^*$ to $\bigl(f(x_n)\bigr)\in\ell_1$. You need to show this is bounded. Use Daniel's hint.
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