See attached image.
I am not sure of how to use the bracketed numbers [0, 1-1/n] for each case.
Any help appreciated,
thanks
Hi again,
I don't mean to come off as rude or ungrateful, as you are offering me help out of your spare time for nothing.
However, I get the feeling you are being vague on purpose with your replies.
If that is not the case, then I apologise, I am somewhat frustrated. If it is the case, can you please stop trying to confuse me?
Thanks.
Anyone of us can only help if you have a basic idea as to the material that you are asking about. Why were you asked to do a problem containing closed intervals if you do not know how they are defined?
$\displaystyle \left[ {0,1 - {n^{ - 1}}} \right] = \left\{ {x:0 \leqslant x \leqslant 1 - {n^{ - 1}}} \right\}$
$\displaystyle \\n=1,~\{0\}\\n=2,~\left[ {0, {\tfrac{1}{2}}}\right]\\n=3,~\left[ {0, {\tfrac{2}{3}}}}\right]\\\cdots$
The problem is you never did say what you did understand about the symbols and what it was that you did not. So usagi has been trying to explain every part. Did you try putting numbers into the formula? When n= 1, [0, 1- 1/n] is [0, 1-1]= [0, 0] which is the "singleton" set, {0}. When n= 2, [0, 1- 1/2]= [0, 1/2] which is the closed interval from 0 to 1/2. When n= 3, [0, 1- 1/3]= [0, 2/3] which is the closed interval from 0 to 2/3, etc.
$\displaystyle \cup_{n=2}^\infty [0, 1- \frac{1}{n}]$
means $\displaystyle [0, 0]\cup [0, 1- \frac{1}{2}]\cup [0, 1- \frac{1}{3}]\cup [0, 1-\frac{1}{4}]\cdot\cdot\cdot$
which is the same as $\displaystyle {0}\cup [0, 1/2]\cup [0, 2/3]\cup [0, 3/4]\cdot\cdot\cdot$
You should be able to see that the intervals always have left endpoint 0 and then go up to some 1- 1/n= n/n- 1/n= (n- 1)/n. As n gets larger and larger that goes to 1 (1/2, 2/3, 3/4, 4/5, ... goes to 1) but is never actually equal to 1 so we have every number from 0 (including 0) up to 1 (but NOT including 1).
$\displaystyle \cup_{n=1}^\infty [0, 1- \frac{1}{n}]= [0, 1)$
(Notice that, although every interval in the union is closed the union itself is not.)
Thanks for your reply, this makes sense.
I was looking at it completely out of perspective and would like to apologise to the user usagi_killer for my outburst who was doing his/her best to help.
I will do my best to be as specific as possible next time.
I really do appreciate this forum. Although, kind of having a bad day today and am very frustrated. (not trying to justify my rage)
Once again sorry, and thank you all
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