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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with yet another aspect of the proof of Proposition 2.3.15 ...Proposition 2.3.15 and its proof read as follows:

View attachment 9073

At the end of the above proof by Sohrab we read the following:

" ... ... The proposition now follows from \(\displaystyle \text{ lim sup } (t_n) \leq e \leq \text{ lim inf } (t_n)\) and Proposition 2.2.39 (f) ... ... "But ... as far as I can tell from Proposition 2.2.39 (f) we require \(\displaystyle \text{ lim sup } (t_n) = e = \text{ lim inf } (t_n)\) ... in order to conclude \(\displaystyle \text{ lim } (t_n) = e\) ...

... but we only have the condition \(\displaystyle \text{ lim sup } (t_n) \leq e \leq \text{ lim inf } (t_n)\) ... ...

So ... how does the proposition 2..3.15 follow ...?

Can someone please clarify the situation above ...

Help will be appreciated ...

Peter

=========================================================================The above post refers to Proposition 2.2.39 ... so I am providing text of the same ... as follows ... :

View attachment 9074

Hope that helps ...

Peter

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with yet another aspect of the proof of Proposition 2.3.15 ...Proposition 2.3.15 and its proof read as follows:

View attachment 9073

At the end of the above proof by Sohrab we read the following:

" ... ... The proposition now follows from \(\displaystyle \text{ lim sup } (t_n) \leq e \leq \text{ lim inf } (t_n)\) and Proposition 2.2.39 (f) ... ... "But ... as far as I can tell from Proposition 2.2.39 (f) we require \(\displaystyle \text{ lim sup } (t_n) = e = \text{ lim inf } (t_n)\) ... in order to conclude \(\displaystyle \text{ lim } (t_n) = e\) ...

... but we only have the condition \(\displaystyle \text{ lim sup } (t_n) \leq e \leq \text{ lim inf } (t_n)\) ... ...

So ... how does the proposition 2..3.15 follow ...?

Can someone please clarify the situation above ...

Help will be appreciated ...

Peter

=========================================================================The above post refers to Proposition 2.2.39 ... so I am providing text of the same ... as follows ... :

View attachment 9074

Hope that helps ...

Peter