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

• Mathematicsresear
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.
Mathematicsresear

## 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

## 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?

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

## 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}##.

## 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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