If E is a set and Y is a point that is the limit of two sequences, <mrow class="MJX-TeXAt

Extrakt04

Extrakt04

Answered question

2022-06-17

If E is a set and Y is a point that is the limit of two sequences, x n and y n such that x n is in E and y n is an upper bound for E, prove that y = sup E. Is the converse true?

Answer & Explanation

Brendon Fernandez

Brendon Fernandez

Beginner2022-06-18Added 14 answers

y n is an upper bound of E
We have to prove that y = sup E
Let sup y n = M, then for given E > 0 there exist no such that M E < y n since y n is the increasing, we get
y n 0 y n
n n 0 implies that
M E y n M M + E , n n 0

This shows y n M
So, y = sup E

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