Analysis Proof: prove that sqrt(x_n) also tends to 0

In summary, the conversation discusses the proof of the statement that if a sequence x_n tends to 0 as n approaches infinity, then sqrt(x_n) also tends to 0. The proof involves choosing a value for epsilon and showing that for all values of x_n greater than N, the square root of x_n is also less than epsilon. One potential issue with the proof is that it does not consider the case where x_n is less than 1.
  • #1
Mathematicsresear
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Homework Statement


Suppose sequence x_n tends to 0 as n approaches infinity, prove that sqrt(x_n) also tends to 0
x_n is a sequence of non negative real numbers

Homework Equations

The Attempt at a Solution


Proof. Let e>0. There exists an N in the naturals such that for n>N Ix_nI < e So if I choose an N such that x_n < e whenever n>N then Isqrt(x_n)I<x_n<e

e is epsilon.

Is this proof correct?
 
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  • #2
Mathematicsresear said:

Homework Statement


Suppose sequence x_n tends to 0 as n approaches infinity, prove that sqrt(x_n) also tends to 0
x_n is a sequence of non negative real numbers

Homework Equations

The Attempt at a Solution


Proof. Let e>0. There exists an N in the naturals such that for n>N Ix_nI < e So if I choose an N such that x_n < e whenever n>N then Isqrt(x_n)I<x_n<e

e is epsilon.

Is this proof correct?
What if ##x_n < 1\,?## E.g. ##\sqrt{\frac{1}{100}} = \frac{1}{10} > \frac{1}{100}##.
 

Related to Analysis Proof: prove that sqrt(x_n) also tends to 0

1. What is an "Analysis Proof"?

An analysis proof is a mathematical method used to prove the validity of a statement or theorem. It involves breaking down a complex problem into smaller, more manageable pieces and using logical reasoning to show that the statement is true.

2. What does it mean for something to "tend to 0"?

When something tends to 0, it means that the value of that thing approaches 0 as its input or variable increases. In other words, the value of the expression gets closer and closer to 0 as the input gets larger and larger.

3. What is the significance of proving that sqrt(x_n) tends to 0?

This proof is significant because it shows that the square root of a sequence of numbers (x_n) also approaches 0 as n increases. This can be applied to various mathematical problems and helps us understand the behavior of certain functions and sequences.

4. What is the process for proving that sqrt(x_n) tends to 0?

The process involves using the definition of a limit, which states that for a function f(x), the limit as x approaches a is L if for any given epsilon, there exists a delta such that when the distance between x and a is less than delta, the distance between f(x) and L is less than epsilon. By applying this definition to the function sqrt(x_n), we can show that it approaches 0 as n increases.

5. Can this proof be applied to other functions or sequences?

Yes, this proof can be applied to various other functions and sequences. The concept of a limit and the process of using a proof by definition can be applied to many different mathematical problems. It is a fundamental concept in calculus and is used to prove many theorems in mathematics.

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