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I am reading Manfred Stoll's book: Introduction to Real Analysis.

I need help with Stoll's proof of Theorem 3.1.16

Stoll's statement of Theorem 3.1.16 and its proof reads as follows:

View attachment 9514Can someone please help me to demonstrate a formal and rigorous proof of the following:If \(\displaystyle U = X \cap O\) for some open subset \(\displaystyle O\) of \(\displaystyle \mathbb{R}\) ...

... then ...

... the subset \(\displaystyle U\) of \(\displaystyle X\) is open in \(\displaystyle X\) ...

Help will be much appreciated ...

My thoughts so far as as follows:

Suppose \(\displaystyle U = X \cap O\) for some open subset \(\displaystyle O\) of \(\displaystyle \mathbb{R}\) ...Need to show \(\displaystyle U\) is open in \(\displaystyle X\) ... that is for every \(\displaystyle p \in U\) there exists \(\displaystyle \epsilon \gt 0\) such that \(\displaystyle N_{ \epsilon } (p) \cap X \subset U\) ... ... Now ... let \(\displaystyle p \in U\) ...

then \(\displaystyle p \in O\) ...

Therefore there exists \(\displaystyle \epsilon \gt 0\) such that \(\displaystyle N_{ \epsilon } (p) \subset O\) ... since \(\displaystyle O\) is open ...BUT ...

... how do I proceed from here ... ?

Hope someone can help ...

Peter

I need help with Stoll's proof of Theorem 3.1.16

Stoll's statement of Theorem 3.1.16 and its proof reads as follows:

View attachment 9514Can someone please help me to demonstrate a formal and rigorous proof of the following:If \(\displaystyle U = X \cap O\) for some open subset \(\displaystyle O\) of \(\displaystyle \mathbb{R}\) ...

... then ...

... the subset \(\displaystyle U\) of \(\displaystyle X\) is open in \(\displaystyle X\) ...

Help will be much appreciated ...

My thoughts so far as as follows:

Suppose \(\displaystyle U = X \cap O\) for some open subset \(\displaystyle O\) of \(\displaystyle \mathbb{R}\) ...Need to show \(\displaystyle U\) is open in \(\displaystyle X\) ... that is for every \(\displaystyle p \in U\) there exists \(\displaystyle \epsilon \gt 0\) such that \(\displaystyle N_{ \epsilon } (p) \cap X \subset U\) ... ... Now ... let \(\displaystyle p \in U\) ...

then \(\displaystyle p \in O\) ...

Therefore there exists \(\displaystyle \epsilon \gt 0\) such that \(\displaystyle N_{ \epsilon } (p) \subset O\) ... since \(\displaystyle O\) is open ...BUT ...

... how do I proceed from here ... ?

Hope someone can help ...

Peter