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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 an another aspect of the proof of Proposition 2.3.15 ...

Proposition 2.3.15 and its proof read as follows:

View attachment 9072

In the above proof by Sohrab, we read the following:

" ... ... It follows that \(\displaystyle t_n \leq s_n\) so that

\(\displaystyle \text{ lim sup } ( t_n ) \leq e\) ... ... "

Can someone please explain exactly how/why \(\displaystyle t_n \leq s_n \Longrightarrow \text{ lim sup } ( t_n ) \leq e\) ... ... ?

My thoughts so far are as follows:

\(\displaystyle t_n \leq s_n\)\(\displaystyle \Longrightarrow \lim_{ n \to \infty } t_n \leq \lim_{ n \to \infty } s_n\) \(\displaystyle \Longrightarrow \lim_{ n \to \infty } t_n \leq e\)But ... how/why can we conclude that \(\displaystyle \text{ lim sup } ( t_n ) \leq e\) ... ... ?

***EDIT*** In the above thoughts I have wrongly assumed that we know, without further analysis, that \(\displaystyle (t_n)\) is convergent ... Help will be appreciated ... ...

Peter

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

I need help with an another aspect of the proof of Proposition 2.3.15 ...

Proposition 2.3.15 and its proof read as follows:

View attachment 9072

In the above proof by Sohrab, we read the following:

" ... ... It follows that \(\displaystyle t_n \leq s_n\) so that

\(\displaystyle \text{ lim sup } ( t_n ) \leq e\) ... ... "

Can someone please explain exactly how/why \(\displaystyle t_n \leq s_n \Longrightarrow \text{ lim sup } ( t_n ) \leq e\) ... ... ?

My thoughts so far are as follows:

\(\displaystyle t_n \leq s_n\)\(\displaystyle \Longrightarrow \lim_{ n \to \infty } t_n \leq \lim_{ n \to \infty } s_n\) \(\displaystyle \Longrightarrow \lim_{ n \to \infty } t_n \leq e\)But ... how/why can we conclude that \(\displaystyle \text{ lim sup } ( t_n ) \leq e\) ... ... ?

***EDIT*** In the above thoughts I have wrongly assumed that we know, without further analysis, that \(\displaystyle (t_n)\) is convergent ... Help will be appreciated ... ...

Peter

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