Am I correct over statements below?
The limsup and liminf of the sequence $n^2$ (meaning $1,4,9,16,\dots$) are equal.
T
Every bounded sequence has at most one convergent subsequence.
F
Are the following characteristic functions Riemann integrable on the interval $[0,1]$?
- $\chi_{\left[0,\frac12\right]}$
yes
- $\chi_{\Bbb Q}$
no
- $\chi_C$, where $C$ is the Cantor set
yes
- $\chi_{\Bbb R-\Bbb Q}$
no
- $\chi_{\left\{\frac1n:n\in\Bbb N\right\}}$
no
- $\chi_{\left[0,\frac12\right]}$
No continuous function $f:\Bbb R\to\Bbb R$ can have a minimum value.
(False)
Let $I_1\supset I_2\supset I_3\supset\dots$ be a nested sequence of closed intervals in $\Bbb R$ whose lengths form a decreasing sequence converging to $0$. Choose points $a_n\in I_n$ for each $n$. Then the sequence $a_n$ converges,
(I think it’s true)
Consider a function $f:\Bbb R\to\Bbb R$. Which of the following statements are true?
- If $f$ is continuous, then it maps every compact set onto a compact set?
yes
- If $f$ maps every compact set onto a compact set, then it is continuous.
no
- If $f$ is continuous, then it maps every connected set onto a connected set?
yes
- Is it true that if $f$ maps every connected set onto a connected set, then it is continuous.
no
- Is it true that if $f$ is continuous, then it maps every open set onto an open set?
yes
- If $f$ maps every open set onto an open set, then it is continuous.
yes
- If $f$ is continuous, then it maps every compact set onto a compact set?
(The original image from which this is copied is here.)