Działania na granicach ciągów

Jeżeli ${\displaystyle \underset{n\to \infty }{\lim }{a}_n=a}$ i ${\displaystyle \underset{n\to \infty }{\lim }{b}_n=b}$ to:

${\displaystyle \underset{n\to \infty }{\lim }(a_n+b_n)=a+b}$,

${\displaystyle \underset{n\to \infty }{\lim }(a_n-b_n)=a-b}$,

${\displaystyle \underset{n\to \infty }{\lim }\left(a_n{b}_n\right)=ab}$,

$\underset{k\in N^{+}}{\exists }\underset{n>k}{\forall }(b_n\ne 0\wedge b\ne 0)\Rightarrow {\displaystyle \underset{n\to \infty }{\lim }\frac{a_n}{b_n}=\frac{a}{b}}$,

$\underset{k\in N^{+}}{\exists }\underset{n>k}{\forall }(a_n\ge b_n\Rightarrow a\ge b)$.

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