3102 said:Homework Statement
I have given the statements: ##a_{n}^2 \ge x## , ##a_{n+1} \le a_{n}## , ##x > 0## and ##\inf a_{n} > 0 ##. How to prove the following: ##\lim_{n \to \infty}a_{n}=\sqrt{x}##Homework Equations
##a_{n}^2 \ge x## , ##a_{n+1} \le a_{n}## , ##x > 0## and ##\inf a_{n} > 0 ##
##\lim_{n \to \infty}a_{n}=\sqrt{x}##
The Attempt at a Solution
I have come so far: $$a_{n}\ge a_{n+1} \ge \sqrt{x}$$ How shall I continue?
A limit in mathematics is a fundamental concept that describes the behavior of a function as its input approaches a particular value. It is used to determine the value that a function approaches as its input gets closer and closer to a specific value, without actually reaching it.
To prove a limit, you must show that the function approaches a specific value as its input approaches a certain value. This can be done using various techniques such as the epsilon-delta definition, the squeeze theorem, or direct substitution.
The epsilon-delta definition of a limit is a mathematical definition used to prove that a function approaches a specific value as its input approaches a given value. It states that for every positive value of epsilon, there exists a corresponding positive value of delta such that when the input is within delta units of the given value, the output is within epsilon units of the desired limit.
The squeeze theorem, also known as the sandwich theorem, is a method used to prove a limit by comparing the given function to two other functions whose limits are known. If the two functions "squeeze" the original function between them, then the limit of the original function must also exist and be equal to the limits of the two comparison functions.
Direct substitution is a method used to evaluate a limit by plugging the given value directly into the function. This method can only be used if the function is continuous at the given value, meaning that the limit exists and is equal to the value of the function at that point.