Check my proof on limit of two sequences

In summary, the conversation discusses a proof involving sequences and limits, where the goal is to show that the limit of the sum of two sequences is equal to the sum of their individual limits. The proof uses the triangle inequality and the fact that for every positive number, there are numbers in the sequences that satisfy certain inequalities. The conversation also mentions the importance of A and B being finite for the proof to be valid.
  • #1
Samuelb88
162
0

Homework Statement


Let [tex]S_n[/tex] and [tex]Q_n[/tex] be sequences and suppose [tex] \lim_{n\rightarrow +\infty} {S_n} = A[/tex] and [tex]\lim_{n\rightarrow +\infty} {Q_n} = B[/tex]. Then [tex] \lim_{n\rightarrow +\infty} {(S_n + Q_n)} = A+B[/tex].

The Attempt at a Solution


*I am using "E" in place of ε.

Proof: I want to show for every E > 0 there is an N such that whenever n>N, [tex]|S_n + Q_n - (A+B)|
< E[/tex].

Suppose [tex] \lim_{n\rightarrow +\infty} {S_n} = A[/tex] and [tex]\lim_{n\rightarrow +\infty} {Q_n} = B[/tex]. Then for every [tex]E_1 > 0[/tex], there are numbers I,J such that whenever:

1. [tex] i > I, |S_i - A| < E_1[/tex]
2. [tex] j > J, |Q_j - B| < E_1[/tex]

Take N=max(I,J) to ensure both the inequalities 1. and 2. will hold. Then by the triangle inequality

[tex]|S_n - A + Q_n -B| <= |S_n - A| + |Q_n - B| < 2E_1[/tex]

Let [tex]E_1 = \frac{1}{2} E[/tex]. Then

[tex]|S_n - A + Q_n -B| < E [/tex] as required. Q.E.D.

Look good?
 
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  • #2
Samuelb88 said:

Homework Statement


Let [tex]S_n[/tex] and [tex]Q_n[/tex] be sequences and suppose [tex] \lim_{n\rightarrow +\infty} {S_n} = A[/tex] and [tex]\lim_{n\rightarrow +\infty} {Q_n} = B[/tex]. Then [tex] \lim_{n\rightarrow +\infty} {(S_n + Q_n)} = A+B[/tex].

The Attempt at a Solution


*I am using "E" in place of ε.

Proof: I want to show for every E > 0 there is an N such that whenever n>N, [tex]|S_n + Q_n - (A+B)|
< E[/tex].

Suppose [tex] \lim_{n\rightarrow +\infty} {S_n} = A[/tex] and [tex]\lim_{n\rightarrow +\infty} {Q_n} = B[/tex]. Then for every [tex]E_1 > 0[/tex], there are numbers I,J such that whenever:

1. [tex] i > I, |S_i - A| < E_1[/tex]
2. [tex] j > J, |Q_j - B| < E_1[/tex]

Take N=max(I,J) to ensure both the inequalities 1. and 2. will hold. Then by the triangle inequality

[tex]|S_n - A + Q_n -B| <= |S_n - A| + |Q_n - B| < 2E_1[/tex]

Let [tex]E_1 = \frac{1}{2} E[/tex]. Then

[tex]|S_n - A + Q_n -B| < E [/tex] as required. Q.E.D.

Look good?

Look's just fine to me.
 
  • #3
This is only true if A and B are finite (i.e not +/-infinity) . If they are then the proof is flawless :).
 
  • #4
╔(σ_σ)╝ said:
This is only true if A and B are finite (i.e not +/-infinity) . If they are then the proof is flawless :).

Woops, forgot to mention that. :) Thanks, guys!
 

FAQ: Check my proof on limit of two sequences

What is the limit of a sequence?

The limit of a sequence is the value that the terms of the sequence approach as the index of the terms increases without bound. In other words, it is the value that the terms "tend towards" as the sequence continues.

How do I check my proof on the limit of two sequences?

To check your proof on the limit of two sequences, you can use the definition of a limit to see if your proof is logically sound. This involves showing that for any positive number ε, there is a corresponding positive integer N such that the terms of the sequences are within ε of the limit for all indices greater than or equal to N.

Can the limit of two sequences be different?

Yes, the limit of two sequences can be different. Each sequence has its own unique behavior and may approach a different limit. However, there are certain conditions under which the limit of two sequences may be the same, such as if the sequences are related or if they have a common factor.

What is the difference between the limit of a sequence and the limit of a function?

The limit of a sequence is the value that the terms of the sequence approach as the index of the terms increases without bound. The limit of a function, on the other hand, is the value that the function approaches as the input approaches a certain value. While both involve approaching a value, they are different concepts and have different definitions.

Why is it important to understand the limit of two sequences?

Understanding the limit of two sequences is important because it allows us to determine the behavior of the sequence and make predictions about its future terms. It also has applications in various fields such as calculus, statistics, and computer science. Additionally, understanding the limit of two sequences helps us to develop a deeper understanding of mathematical concepts and proofs.

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