Sequence (a_n)^2 ->0 implies (a_n) ->0 ?

• kingwinner
In summary: Otherwise, you may need to find an N depending on epsilon. In summary, the question is whether an sequence converges to 0 as n approaches infinity. The answer is yes, based on the definition of convergence. Furthermore, if an sequence converges to 0, its square also converges to 0. This can be proved using the concept of continuity and epsilon-delta proofs.
kingwinner

Homework Statement

Let an be an infinite sequence.

1) Is it true that an2 ->0 as n->∞
=> an ->0 as n->∞ ?

2) Is it also true that an ->0 as n->∞
=> an2 ->0 as n->∞ ?

N/A

The Attempt at a Solution

Both SEEM to be true to me, but I am not sure why. What is the simplest way to explain these? I want to understand intuitively.

Any help is appreciated!

Saying that $a_n\to 0$ means that, given any $\epsilon> 0$ there exist N such that if n> N, $|a_n- 0|= |a_n|< \epsilon$.

Saying that $a_n^2\to 0$ means that, given any $\epsilon> 0$ there exist N such that if n> N, $|a_n^2- 0|= |a_n^2|< \epsilon$.

If you know that $a_n\to 0$ then, given any $\epsilon> 0$ there exist N such that if n> N, $|a_n|< \sqrt{\epsilon}$. From that it follows that, for n> N, $|a_n^2|< \epsilon$.

Conversely, if you know that [itxex]a_n^2\to 0[/itex] then, given any $\epsilon> 0$, there exist N such that if n> N, $|a_n^2|< \epsilon^2$. From that it follows that, for n> N, $|a_n|< \epsilon$.

More generally, if $a_n\to a$ and f(x) is continuous in some neighborhood of a, then $f(a_n)\to f(a)$.

HallsofIvy said:
More generally, if $a_n\to a$ and f(x) is continuous in some neighborhood of a, then $f(a_n)\to f(a)$.

Thanks! But I have a question, then.
an2 ->0 as n->∞
=> an ->0 as n->∞

But the square root function is NOT continuous at 0, and it's definitely NOT continuous in a neighbourhood of 0, so the theorem does not apply??
It's weird...

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Halls said 'more generally'. sqrt is one-sided continuous at 0, from the positive side. If you don't have special theorems about one sided continuity, then just pay attention to his suggestions about an epsilon delta proof. Isn't that what you meant about "SEEM to be true"?

I have a question about the proof.

Saying that $a_n\to 0$ means that, given ANY $\epsilon> 0$ there exist N such that if n> N, $|a_n- 0|= |a_n|< \epsilon$.

Because of the word "ANY", we can take epsilon here to be $\sqrt{\epsilon}$, and there would still exist some N, right?

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You can if you know as epsilon approaches zero, sqrt(epsilon) approaches zero. If you are ok with that, go for it.

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1. What is the significance of the sequence (a_n)^2 approaching 0?

The sequence (a_n)^2 approaching 0 indicates that the terms in the sequence are decreasing in magnitude, which suggests that the terms in the original sequence (a_n) are also decreasing. This is important because it helps prove that the sequence (a_n) is convergent, meaning it approaches a specific limit as n approaches infinity.

2. How does the behavior of (a_n)^2 affect the behavior of (a_n)?

Since (a_n)^2 is a square of the terms in (a_n), it is always positive. This means that if (a_n)^2 approaches 0, then (a_n) must also approach 0. This is because for any positive number a, the square root of a (denoted as √a) will also be positive. Therefore, as (a_n)^2 approaches 0, the corresponding √(a_n)^2 will also approach 0.

3. Can (a_n)^2 approach 0 while (a_n) does not?

No, this is not possible. As mentioned earlier, the square root of any positive number will always be positive. Therefore, if (a_n)^2 approaches 0, then (a_n) must also approach 0.

4. Is the converse of this statement true?

No, the converse of this statement is not true. Just because a sequence (a_n) approaches 0 does not necessarily mean that its square (a_n)^2 will also approach 0. This is because (a_n)^2 can still approach 0 even if (a_n) does not approach 0, as long as (a_n) remains bounded.

5. How is this theorem useful in real-world applications?

This theorem is useful in many real-world applications, particularly in fields such as physics and engineering. It helps to prove the convergence of certain mathematical models and predictions. For example, in physics, this theorem can be applied to analyze the behavior of objects in motion or the decay of radioactive materials. In engineering, it can be used to ensure the stability and accuracy of numerical methods and simulations.

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