All Questions
7
questions
0
votes
4
answers
67
views
Rudin PMA 4.31 - does the elements of $E$ have to be ordered (smallest to the biggest)?
Here is a reformulation of Rudin PMA $4.31$ remark:
Let $a$ and $b$ be two real numbers such that $a < b$, let $E$ be any countable subset of the open interval $(a,b)$, and let the elements of $E$ ...
- 1,231
2
votes
2
answers
123
views
Formula for the $n$-th term of the sequence $1, 2, 4, 6, 12, 16, 24, 30, 60, 72, 96, 112, \ldots$, where $f_n := \frac{1}{n} (f_{2n} - f_n)$
I'm struggling with this sequence.
$$1, 2, 4, 6, 12, 16, 24, 30, 60, 72, 96, 112, \ldots$$
Where, $f_n := \frac{1}{n} (f_{2n} - f_n)$
You can also work it out for negative powers of 2,
$$f_\frac{1}{2} ...
- 71
0
votes
0
answers
124
views
Converse of Bolzano Weierstrass Theorem
Bolzano Weierstrass Theorem (for sequences) states that Every bounded sequence has a limit point.
However the converse is not true i.e. there do exist unbounded sequence(s) having only one real limit ...
0
votes
0
answers
38
views
Finding Sequence of Polynomials Whose Existence is Guaranteed
I'm interested in knowing whether we can find a sequence of polynomials (thanks to Stone-Weierstras) that converges to the Weierstrass function on some random interval (for instance, [-2 , 2]. The ...
- 13
3
votes
1
answer
81
views
Suppose $\sum_{n\ge 1} |a_n| = A<\infty.$ Under what conditions is $\sum_{n\ge 1} \epsilon_n a_n = [-A,A]$, for $\epsilon_n \in \{-1,1\}$?
Consider the space of sequences:
$$
\mathcal{E} = \{\{\epsilon_n\}_{n= 1}^{\infty}: \epsilon_n = \pm 1\}
$$
This can be considered a "random choice of sign" in the probabilistic context, for ...
- 8,369
1
vote
1
answer
222
views
Show function does not have limit using sequence.
Consider the function g: $R$ $ \rightarrow $ $R$ defined by
$$
g(x) = \left\{\begin{aligned}
&4-.5x &&: x\,rational\\
&.5x &&: x\,irrational
\end{aligned}
\right.$$
Pick a ...
- 31
2
votes
1
answer
448
views
construct a sequences of integrable function that tends to the dirichlet function.
so I wanted to ask if (it is even possible) to construct a sequence of integrable function $f_n$ such that $f_n \rightarrow f$ where $f$ is the dirichlet function.
$f := \begin{cases}0\ \ x\in[a,b]\...
- 2,157