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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 $$t_n \leq s_n$$ so that
$$\text{ lim sup } ( t_n ) \leq e$$ ... ... "
Can someone please explain exactly how/why $$t_n \leq s_n \Longrightarrow \text{ lim sup } ( t_n ) \leq e$$ ... ... ?
My thoughts so far are as follows:
$$t_n \leq s_n$$$$\Longrightarrow \lim_{ n \to \infty } t_n \leq \lim_{ n \to \infty } s_n$$ $$\Longrightarrow \lim_{ n \to \infty } t_n \leq e$$But ... how/why can we conclude that $$\text{ lim sup } ( t_n ) \leq e$$ ... ... ?
***EDIT*** In the above thoughts I have wrongly assumed that we know, without further analysis, that $$(t_n)$$ 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 $$t_n \leq s_n$$ so that
$$\text{ lim sup } ( t_n ) \leq e$$ ... ... "
Can someone please explain exactly how/why $$t_n \leq s_n \Longrightarrow \text{ lim sup } ( t_n ) \leq e$$ ... ... ?
My thoughts so far are as follows:
$$t_n \leq s_n$$$$\Longrightarrow \lim_{ n \to \infty } t_n \leq \lim_{ n \to \infty } s_n$$ $$\Longrightarrow \lim_{ n \to \infty } t_n \leq e$$But ... how/why can we conclude that $$\text{ lim sup } ( t_n ) \leq e$$ ... ... ?
***EDIT*** In the above thoughts I have wrongly assumed that we know, without further analysis, that $$(t_n)$$ is convergent ... Help will be appreciated ... ...
Peter
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