Rules for convergent sequences
Let
and
be
convergent sequences. Then the following statements hold.
- The sequence
is convergent, and
-
![{\displaystyle {}\lim _{n\rightarrow \infty }{\left(x_{n}+y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}+{\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccaea0883f278be4e8840523b29c361eeaeffbd0)
holds.
- The sequence
is convergent, and
-
![{\displaystyle {}\lim _{n\rightarrow \infty }{\left(x_{n}\cdot y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}\cdot {\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fef7fa4d521a34a759b61acd5c5d3e9ae86fa4dd)
holds.
- For
,
we have
-
![{\displaystyle {}\lim _{n\rightarrow \infty }cx_{n}=c{\left(\lim _{n\rightarrow \infty }x_{n}\right)}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c51f600686e16716d9a442c06d9a5aa6f05a107e)
- Suppose that
and
for all
.
Then
is also convergent, and
-
![{\displaystyle {}\lim _{n\rightarrow \infty }{\frac {1}{x_{n}}}={\frac {1}{x}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9638896a75aaf6719bde39319af296495efd137)
holds.
- Suppose that
and that
for all
.
Then
is also convergent, and
-
![{\displaystyle {}\lim _{n\rightarrow \infty }{\frac {y_{n}}{x_{n}}}={\frac {\lim _{n\rightarrow \infty }y_{n}}{x}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94a7abb2a5d29b25c5885ca57802fbe26fd6bc9d)
holds.