I need to find:
$\lim_{n\to \infty} a_n =\lim_{n\to \infty} \frac{1}{n^2 +n} , for: \forall n \in \Bbb N \setminus \{ 0 \}$
By using the Sandwich a.k.a. squeeze theorem.My ideas so far:
when i factor out the n, i get:
$a_n = \frac{1}{n(n +1)}$
from that i can build the following inequations:
1: $0< \frac{1}{n} \le 1 , for: \forall n \in \Bbb N \setminus \{ 0 \}$
2: $0< \frac{1}{n+1} \le \frac{1}{2} , for: \forall n \in \Bbb N \setminus \{ 0 \}$
now when i multiply first inequality with $\frac{1}{n+1}$ i will get the following:
$0\left( \frac{1}{n+1} \right) < \frac{1}{n(n +1)} \le \left( \frac{1}{n+1} \right)1 $
or if i multiply the second inequality with $\frac1n$ i will get:
$0\left( \frac1n \right) < \frac{1}{n(n +1)} \le \left( \frac1n \right)1 $
My Problem with this approach:
The leftmost side of this inequality is smaller but not smaller or equal then the middle sequence, but i don't know if this is valid, and i dont know if one can simply say $\lim_{n \to \infty} 0 = 0$ either.
The second Problem is i would like to know if i choose the flanking sequences the right way.