A = (a,b)
B = [a,b)
C = [a,b]
D = (a,b]
, a,b are real numbers
Which one of these sets is the biggest?
(Proof is not necessary, I only need to know the correct answer)
Ok, thank you.
Another question:
I have prooved that a function converges in all these 4 sets.
I would just like to use the "biggest" of them to show that it converges in the a larger area (like trying to include all the 4 of them, in a sentence) and I am wondering which one to use.
Maybe the set "C" ?
(assuming , $\displaystyle a<b$...)
Here is a hint. Define $\displaystyle f:[0,1]\to (0,1)$ by $\displaystyle f(0)=\frac{1}{2},~ f(1)=\frac{1}{3}, f(1/n)=\frac{1}{n+1}, n\ge3,~x\text{ otherwise} $.
It is easy to show that $\displaystyle f$ is a bijection.
Thus we see that $\displaystyle [0,1]~\&~(0,1)$ are ‘same size’.
In this case what matters is being the supper-set.
That is, if I understand what you mean by “a function converges”.
For example: If $\displaystyle f$ is continuous on $\displaystyle [a,b]$ then it is continuous on $\displaystyle (a,b)$.
"Bigger than" has not meaning in this context.