Which of the following statements are true and why?
Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.
$\log x$ is uniformly continuous on $( 1/2,+\infty)$.
If $A, B$ are closed subsets of $[0,\infty)\,$, then $A + B = \{x + y\; |\; x \in A,\, y \in B\}$ is closed in $[0,\infty)$
A bounded continuous function on $\mathbb{R}$ is uniformly continuous.
Suppose $f_n(x)$ is a sequence of continuous functions on the closed interval $[0, 1]$ converging to $0$ pointwise. Then the integral $\int_0^1f_n(x)\mathrm dx\,$ converges to $0$.
My thoughts:
- I am not sure asthe interval is not closed.
- It is true as it has bounded derivative.
- Usually, the sum of two closed set is not closed but is the case here?
- Not sure.
- Not sure.
Can anyone help me please to solve the problems? Thank you.