The book proves this limit and I am a bit confused how all the pieces fit together.(adsbygoogle = window.adsbygoogle || []).push({});

So the book proves "If [itex](s_n)[/itex] converges to [itex]s[/itex] and [itex](t_n)[/itex] converges to [itex]t[/itex], then [itex](s_nt_n)[/itex] converges to [itex]st [/itex]. That is, [itex] lim(s_nt_n) = (lim s_n)(lim t_n)[/itex].

The proof goes like this

Let [itex] \epsilon> 0 [/itex] . By Theorem 9.1 there is a constant [itex] M > 0[/itex] such that

[itex]|s_n| ≤ M [/itex]for all [itex]n[/itex]. Since [itex]lim t_n = t[/itex] there exists [itex]N_1 [/itex] such that [itex]n > N1 [/itex] implies [itex]|t_n − t| <\epsilon/(2M) [/itex] Also, since [itex] lim s_n = s [/itex] there exists [itex]N_2 [/itex] such that [itex]n > N_2 [/itex]implies [itex]|s_n − s| < \epsilon/(2(|t| + 1)) [/itex] Then [itex]|s_nt_n − st| ≤ |s_n| · |t_n − t| + |t| · |s_n − s|

≤ M · (\epsilon/2M)+ |t| · (\epsilon/(2(|t| + 1))<\epsilon/2+\epsilon/2=\epsilon [/itex].

The part I do not understand about the proof is this jump in the inequality in the last step that is how is [itex]|s_n| · |t_n − t| + |t| · |s_n − s|≤ M · (\epsilon/(2M))+ |t| · (\epsilon/(2(|t| + 1))[/itex] instead of just less than?

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# Proof about a limit property clarification

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