MHB Proposition 2.3.15: Understanding Sohrab's Basic Real Analysis Proof

[Math Amateur]

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

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

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

 

[Opalg]

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

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

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

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

It is always true that $\liminf (t_n) \leqslant \limsup(t_n)$. So in the inequalities $ \limsup(t_n) \leqslant e \leqslant \liminf (t_n) \leqslant \limsup(t_n)$ the last term is the same as the first, and therefore there must be equality throughout.

 

[Math Amateur]

It is always true that $\liminf (t_n) \leqslant \limsup(t_n)$. So in the inequalities $ \limsup(t_n) \leqslant e \leqslant \liminf (t_n) \leqslant \limsup(t_n)$ the last term is the same as the first, and therefore there must be equality throughout.

Thanks for the help, Opalg ...

Peter