The central limit theorem, also known as the central limit theorem of probability, states that for any sequence of random variables, the sum of their standard deviations tends to a limiting value as the sample size increases. This theorem is widely used in statistical inference and is fundamental to many areas of mathematics, including calculus, probability theory, and statistics.

Proof of the central limit theorem:

Let X1, X2,..., Xn be a sample of n random variables, each with a normal distribution N(mu, sigma^2). The Central Limit Theorem states that as the sample size n increases, the sum of the squared standard deviations of the random variables tends to a limiting value as sigma^2.

Letting S = sum(X1^2, X2^2,..., Xn^2), we have:

S = ∑(X1^2 + X2^2 +... + Xn^2)

This is because the sum of the squared standard deviations is a sum of the squared standard deviations, and the sum of the squared standard deviations is a continuous function of the sample size n.

Now, let's consider the limit of the sum of the squared standard deviations as the sample size n increases. This limit is denoted by ∞.

Since the standard deviation of each random variable is a continuous function of the sample size n, we can apply the Central Limit Theorem to each random variable separately. That is, we can find the limiting value of the sum of the squared standard deviations for each random variable.

For example, let X1 = N(mu = 0, sigma^2 = 1), X2 = N(mu = 1, sigma^2 = 1),..., Xn = N(mu = 0, sigma^2 = 1).

Using the Central Limit Theorem, we have:

S = ∑(X1^2 + X2^2 +... + Xn^2)

= ∑(X1^2 + X2^2 +... + Xn^2) + ∑(X1^2 + X2^2 +... + Xn^2)

= ∑(X1^2 + X2^2 +... + Xn^2) + ∑(X1^2 + X2^2 +... + Xn^2)

= ∑(X1^2 + X2^2 +... + Xn^2) + ∑(X1^2 + X2^2 +... + Xn^2)

= ∑(X1^2 + X2^2 +... + Xn^2) + ∑(X1^2 + X2^2 +... + Xn^2)

= ∑(X1^2 + X2^2 +... + Xn^2)

Since the sum of the squared standard deviations is a continuous function of the sample size n, we can find the limiting value of the sum of the squared standard deviations for each random variable.

Therefore, the sum of the squared standard deviations for each random variable tends to a limiting value as the sample size n increases, which is the Central Limit Theorem.
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