Beispiele. 2. Sei

$(a_{n}) = (\frac{2 n^{2}}{n^{2} + 2 n + 3}) .$ Behauptung:

$a_{n} \to 2 .$ Beweis. Sei

$ε > 0 .$ Es ist

$\| a_{n} - 2 \|$	=	$\| \frac{2 n^{2}}{n^{2} + 2 n + 3} - 2 \| = \| \frac{2 n^{2} - 2 n^{2} - 4 n - 6}{n^{2} + 2 n + 3} \|$
	=	$\| \frac{- 4 n - 6}{n^{2} + 2 n + 3} \| = \frac{4 n + 6}{n^{2} + 2 n + 3}$
	=	$\underset{\leq \frac{4 n}{n^{2}}}{\underset{︸}{\frac{4 n}{n^{2} + 2 n + 3}}} + \underset{\leq \frac{6}{2 n}}{\underset{︸}{\frac{6}{n^{2} + 2 n + 3}}}$
	$\leq$	$\frac{4}{n} + \frac{3}{n} = \frac{7}{n} .$