Скачать или смотреть 3. Theorem 3.3 | Show that lim_(n→∞)⁡(s_n+t_n )=s+t, lim_(n→∞) ⁡s_n t_n=st | Rudin Real Analysis

Principles of Mathematical Analysis | Real Analysis | Walter Rudin | Chapter 3 Numerical Sequences and series
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This lecture explains Theorem 3.3 from Chapter 3 of Principles of Mathematical Analysis by Walter Rudin. The theorem describes important properties of convergent sequences that can be stated as:
Suppose that {s_n } and {t_n } are complex sequences, and lim_(n→∞)⁡ s_n=s and lim_(n→∞)⁡t_n=t. Then
lim_(n→∞)⁡〖(s_n+t_n )=s+t〗
lim_(n→∞)⁡〖cs_n=cs,〗 lim_(n→∞)⁡〖(c〖+s〗_n )=c+s,〗 for any c;
lim_(n→∞)⁡〖s_n t_n=st〗;
lim_(n→∞)⁡〖1/s_n =1/s〗, provided s_n≠0(n=1,2,3,…) and s≠0.
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