TFAE proof involving limit and convergent sequence

In summary, if limx→∞ f(x) = L, then for every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn)) converges to L, and vice versa.
  • #1
brooklysuse
4
0
Let A ⊆ R, let f : A → R, and suppose that (a,∞) ⊆ A for some a ∈ R. Then the
following statements are equivalent:
i) limx→∞ f(x) = L
ii) For every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn))
converges to L.

Not even sure how to begin this one, other than the fact that proving i) -->ii) and ii)--> i) will be sufficient. Could anyone help me with these 2 parts?

Thanks!
 
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  • #2
For any proof you start by thinking about what the "hypotheses" and "conclusion" mean: the specific definitions.

For (i)-> (ii) we have the hypothesis [tex]\lim_{x\to \infty} f(x)= L[/tex] so start with the definition of that: Given any [tex]\epsilon> 0[/tex] there exist K such that if x> K then [tex]|f(x)- L[/tex]. And the definition of the conclusion, "for any sequence [tex]\{a_m\}[/tex] in [tex]A\cap (a, \infty)[/tex], such that [tex]\lim a_n=\infty[tex], [tex]\lim f(a_n)= L[/tex]": Given any [tex]\epsilon> 0[/tex] there exist N such that if n> N then [tex]|f(a_n)- L|< \epsilon[/tex].

Now combine the two: Given any [tex]\epsilon> 0[/tex], let [tex]\{a_n\}[/tex] be a sequence in [tex]A\cap (a, \infty)[/tex] such that [tex]\lim a_n= \infty[/tex]. From the hypothesis, there exist K such that if x> K, [tex]|f(x)- L|< \epsilon[/tex]. By the definition of "[tex]\lim a_n= \infty[/tex], there exist N such that if n> N, [tex]a_n> K[/tex]. So there exist N such that if n> N [tex]|f(a_n)- L|< \epsilon[/tex] and we are done. (ii)-> (i) is done similarly.

Important point: definitions in mathematics are "working" definitions- you use the precise wording of definitions in proofs.
 
  • #3


Sure, here's how you can approach this proof:

i) --> ii):

Assume that limx→∞ f(x) = L. This means that for any ε > 0, there exists a number M such that for all x > M, |f(x) - L| < ε. Now, consider a sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞. This means that for any M, there exists an N such that for all n > N, xn > M. Since (a,∞) ⊆ A, this also means that for all n > N, xn ∈ A. Therefore, for all n > N, we have xn > M and xn ∈ A, so xn ∈ A ∩ (a,∞). Now, using the definition of limit, we can say that for any ε > 0, there exists an N such that for all n > N, |xn - ∞| < ε. Since xn > M for all n > N, this means that for all n > N, xn > M and |f(xn) - L| < ε. Therefore, by the definition of convergence, we can say that lim(f(xn)) = L.

ii) --> i):

Assume that for every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn)) converges to L. This means that for any ε > 0, there exists an N such that for all n > N, |f(xn) - L| < ε. Now, consider any x > a. Since (a,∞) ⊆ A, this means that x ∈ A. Now, consider a sequence (xn) in A ∩ (a,∞) such that xn = x + n. This sequence satisfies the condition that lim(xn) = ∞. Therefore, by our assumption, we can say that lim(f(xn)) = L. But since xn = x + n, this means that for all n > N, |f(x + n) - L| < ε. Now, using the definition of limit, we can say that limx→∞ f(x) = L.

Therefore, we have shown that i) and ii) are equivalent.
 

Related to TFAE proof involving limit and convergent sequence

1. What is a TFAE proof involving limit and convergent sequence?

A TFAE (The Following Are Equivalent) proof is a type of mathematical proof that shows that multiple statements are equivalent, meaning that they are all true or all false. In this case, the proof involves showing that a given limit and convergent sequence are equivalent.

2. How is a TFAE proof involving limit and convergent sequence useful?

This type of proof is useful because it allows us to establish a relationship between a limit and a convergent sequence. It can also help us to simplify complex mathematical problems and make them easier to solve.

3. What is a limit in mathematics?

In mathematics, a limit is a value that a function or sequence approaches as the input or index approaches a certain value. It is used to describe the behavior of a function or sequence near a particular point or infinity.

4. What is a convergent sequence?

A convergent sequence is a sequence of numbers that approaches a specific value as the index increases. In other words, as the terms in the sequence get closer and closer to each other, they also get closer and closer to a specific value, known as the limit.

5. What is the process for proving a TFAE involving limit and convergent sequence?

The process for proving a TFAE involving limit and convergent sequence typically involves showing that the statements are logically equivalent, using mathematical operations and properties to manipulate the statements and arrive at the same conclusion. It is important to carefully justify each step and use proper notation to clearly communicate the proof.

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