# Category:Limits of Sequences

This category contains results about Limits of Sequences.

Definitions specific to this category can be found in Definitions/Limits of Sequences.

### Real Numbers

Let $\sequence {x_n}$ be a sequence in $\R$.

Let $\sequence {x_n}$ converge to a value $l \in \R$.

Then $l$ is a **limit of $\sequence {x_n}$ as $n$ tends to infinity**.

### Rational Numbers

Let $\sequence {x_n}$ be a sequence in $\Q$.

Let $\sequence {x_n}$ converge to a value $l \in \R$, where $\R$ denotes the set of real numbers.

Then $l$ is a **limit of $\sequence {x_n}$ as $n$ tends to infinity**.

### Complex Numbers

Let $\sequence {z_n}$ be a sequence in $\C$.

Let $\sequence {z_n}$ converge to a value $l \in \C$.

Then $l$ is a **limit of $\sequence {z_n}$ as $n$ tends to infinity**.

## Subcategories

This category has the following 19 subcategories, out of 19 total.

### B

- Bolzano-Weierstrass Theorem (5 P)

### C

### D

### L

### M

### P

- Peak Point Lemma (3 P)

### R

- Reciprocal of Null Sequence (2 P)

### S

## Pages in category "Limits of Sequences"

The following 91 pages are in this category, out of 91 total.

### B

### C

- Cesàro Mean
- Combination Theorem for Sequences
- Combination Theorem for Sequences/Real/Product Rule
- Continuity of Mapping between Metric Spaces by Convergent Sequence
- Convergence by Multiple of Error Term
- Convergence of Limsup and Liminf
- Convergent Real Sequence has Unique Limit
- Convergent Real Sequence is Bounded
- Convergent Sequence has Unique Limit
- Convergent Sequence in Hausdorff Space has Unique Limit
- Convergent Sequence in Metric Space has Unique Limit
- Convergent Sequence in Metric Space is Bounded
- Convergent Sequence in Normed Vector Space has Unique Limit
- Convergent Sequence Minus Limit
- Convergent Subsequence in Closed Interval

### E

### L

- Limit of Bounded Convergent Sequence is Bounded
- Limit of Complex Function by Convergent Sequences
- Limit of Function by Convergent Sequences
- Limit of Function by Convergent Sequences/Complex Plane
- Limit of Function by Convergent Sequences/Corollary
- Limit of Function by Convergent Sequences/Real Number Line
- Limit of Image of Sequence
- Limit of Image of Sequence/Real Number Line
- Limit of Integer to Reciprocal Power
- Limit of Real Function by Convergent Sequences
- Limit of Root of Positive Real Number
- Limit of Sequence in Metric Space in Neighborhood
- Limit of Sequence in Product of Metric Spaces under Chebyshev Distance
- Limit of Sequence to Zero Distance Point
- Limit of Subsequence equals Limit of Sequence
- Limit of Subsequence equals Limit of Sequence/Metric Space
- Limit of Subsequence equals Limit of Sequence/Normed Vector Space
- Limit of Subsequence of Bounded Sequence
- Limit Point is Limit of Convergent Sequence
- Limits Preserve Inequalities
- Limsup and Liminf are Limits of Bounds
- Linear Combination of Sequences
- Lower and Upper Bounds for Sequences
- Lower and Upper Bounds for Sequences/Corollary
- Lower and Upper Bounds for Sequences/Warning

### M

### R

- Rational Sequence Decreasing to Real Number
- Rational Sequence Increasing to Real Number
- Real Bounded Monotone Sequence is Convergent
- Real Bounded Sequence has Convergent Subsequence
- Real Sequence has Monotone Subsequence
- Real Sequence with all Subsequences having Convergent Subsequence to Limit Converges to Same Limit
- Reciprocal of Null Sequence

### S

- Sequence Converges to Within Half Limit
- Sequence Converges to Within Half Limit/Complex Numbers
- Sequence Converges to Within Half Limit/Normed Division Ring
- Sequence Converges to Within Half Limit/Real Numbers
- Sequence of Powers of Number less than One
- Sequence of Powers of Number less than One/Complex Numbers
- Sequence of Powers of Number less than One/Necessary Condition
- Sequence of Powers of Number less than One/Normed Division Ring
- Sequence of Powers of Number less than One/Rational Numbers
- Sequence of Powers of Number less than One/Sufficient Condition
- Sequence of Powers of Reciprocals is Null Sequence
- Sequence of Reciprocals is Null Sequence
- Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals
- Spanish Hotel Theorem
- Squeeze Theorem
- Squeeze Theorem for Sequences of Complex Numbers
- Squeeze Theorem/Sequences
- Squeeze Theorem/Sequences/Complex Numbers
- Squeeze Theorem/Sequences/Linearly Ordered Space
- Squeeze Theorem/Sequences/Metric Spaces
- Stolz-Cesàro Theorem
- Stolz-Cesàro Theorem/Corollary
- Subsequence of Real Sequence Diverging to Negative Infinity Diverges to Negative Infinity
- Subsequence of Real Sequence Diverging to Positive Infinity Diverges to Positive Infinity
- Subsequence of Sequence in Metric Space with Limit