Sequence Limit (abstract math)

In summary, a sequence limit in abstract math is a value that a sequence of numbers approaches as the number of terms in the sequence increases. It is calculated by analyzing the pattern of terms in the sequence and has significance in understanding and predicting the behavior of a sequence. A sequence can only have one limit and a well-defined limit is necessary for a sequence to be considered convergent.
  • #1
FourierX
73
0

Homework Statement



If x_n[tex]\rightarrow[/tex] L and y_n [tex]\rightarrow[/tex] M

prove that

x_n - y_n [tex]\rightarrow[/tex] L-M

Homework Equations



Definition of Limit

The Attempt at a Solution



I followed and stayed within the definition of limit of a sequence, but I got 0 for x_n - y_n.
 
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  • #2
Can you show us what you did then? Probably your approach is correct but you made a small mistake somewhere.
 
  • #3
[itex]|(x_n- y_n)- (L- M)|= |(x_n- L)+ (-y_n+ M)|\le |x_n-L|+ |-y_n+ M|[/itex]
 

1. What is a sequence limit in abstract math?

A sequence limit in abstract math is a value that a sequence of numbers approaches as the number of terms in the sequence increases. It is denoted by the symbol "lim" and is used to analyze the behavior of a sequence as it approaches infinity.

2. How is a sequence limit calculated?

A sequence limit can be calculated by looking at the pattern of the terms in the sequence and finding the value that the terms are approaching as the number of terms increases. This can be done by using various mathematical techniques, such as algebraic manipulation or graphical analysis.

3. What is the significance of a sequence limit in abstract math?

A sequence limit is significant because it helps us understand the behavior of a sequence as it approaches infinity. It also allows us to make predictions about the behavior of a sequence and can be used in various mathematical applications, such as in calculus and statistics.

4. Can a sequence have multiple limits?

No, a sequence can only have one limit. If a sequence has multiple limits, it is considered divergent and does not have a well-defined limit.

5. How is a sequence limit related to the concept of convergence?

A sequence limit is closely related to the concept of convergence, as a sequence is said to converge if it has a well-defined limit. A sequence can also be classified as convergent or divergent based on the behavior of its limit. A convergent sequence has a finite limit, while a divergent sequence does not have a limit or has a limit of infinity.

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