Section 12

Letting $(s_n)$ be the sequence of real numbers and $S$ being the subsequential limits of $S$. Then remember $\lim \sup s_n={\lim }_{N\to \infty }\sup \{s_n:n>N\}=\sup S$ $\lim \inf s_n={\lim }_{N\to \infty }\inf \{s_n:n>N\}=\inf S$

Theorem 1

Suppose $(s_n)$ converges to a positive real number $s$ and $(t_n)$ is any sequence. Then $\lim \sup s_n{t}_n=s\lim \sup t_n$

Theorem 2

Useful for infinite series:

Implication