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"Let A = [0,1[. We prove that A does not have a maximum.

"Let A = [0,1[. We prove that A does not have a maximum.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I have the following example in my analysis textbook 

"Let A = [0,1[. We prove that A does not have a maximum. Suppose A has a maximum M. Then M ∈ A, so 0 ≤ M < 1. Therefore, there exists an x ∈ R such that M < x < 1. For this, we have 0 ≤ M < x < 1, which means x ∈ [0,1[ = A and x > M, contradiction."

I dont really follow it from 'Therefore', how do we know that there exists an 'x' such that 'M < x < 1'?

If able please explain it in a bit more detail, thank you in advance.

 

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