Prove that if m = sup S and m is not an element of S, then m is an accumulation point

warwick said:
Prove that if m = sup S and m is not an element of S, then m is an accumulation point of S and S is an infinite set.
Do you buy my proof?
Click to expand...
Sorry to tell you, but I simply cannot read the image in that link.
If \(\displaystyle m=\sup(M)\) then \(\displaystyle m-1\) is not.
So \(\displaystyle \left( {\exists s_1 \in S} \right)\left[ {m - 1 < s_1 < m} \right]\).
But because \(\displaystyle s_1< m\) then \(\displaystyle \left( {\exists s_2 \in S} \right)\left[ {s_1 < s_2 < m} \right]\).

You can see the if an infinite set of increasing points in \(\displaystyle S\).
 
pka said:
Sorry to tell you, but I simply cannot read the image in that link.
If \(\displaystyle m=\sup(M)\) then \(\displaystyle m-1\) is not.
So \(\displaystyle \left( {\exists s_1 \in S} \right)\left[ {m - 1 < s_1 < m} \right]\).
But because \(\displaystyle s_1< m\) then \(\displaystyle \left( {\exists s_2 \in S} \right)\left[ {s_1 < s_2 < m} \right]\).

You can see the if an infinite set of increasing points in \(\displaystyle S\).
Click to expand...

The image should enlarge enough for you to make out the characters.
 
warwick said:
The image should enlarge enough for you to make out the characters.
Click to expand...
That does not help.
The image is dim and the hand writing is hard to read.
Why not type it out? Learn to use LaTeX.
 
You'll need to explain your notation. N*(m,epsilon) is what?
 
I think you need to clarify the implication that there exists an s with m > s > m' => N*(m,e) intersect S =/= {}. You want to show every neighborhood of m contains a point, correct? So Let N be a neighborhood of m. Find an s in S with this property. And I'm not sure what S' is either.

Your second part confuses me. You assert "there is an s in S for every epsilon-neighborhood of m", is that what you meant to say? This statement implies "the number of elements in S is at least as large as the number of neighborhoods of m"... which is not true in general. Try to write your proof clearly. Mathematics is written in sentences, so if you must use symbols, be sure everything reads like it should.
 
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