Epsilon-Delta Proof

Epsilon-Delta proof is a method of proving mathematical statements related to limits, continuity, and convergence. It is an important concept in calculus and real analysis.

The essence of an Epsilon-Delta proof is to show that a given limit exists, by finding a value of delta that corresponds to a given epsilon.

An Epsilon-Delta proof can be written in the form of:

For all ε > 0, there exists a δ > 0 such that |x - c| < δ implies |f(x) - L| < ε.

Where:

• ε is a small positive number, representing the allowable error or tolerance.
• δ is a small positive number, representing the distance from the point c over which the function f(x) is to be examined.
• c is the limit point where x is approaching.
• L is the limit of f(x) as x approaches c.

Let's take an example of an Epsilon-Delta proof for the limit of a function f(x) = 2x + 3, as x approaches 2.

## Example Epsilon-Delta Proof

We want to show that the limit of the function f(x) = 2x + 3, as x approaches 2, is 7.

We start by assuming ε > 0, and we want to find a δ that satisfies

| x - 2 | < δ => | (2x + 3) - 7 | < ε.

To simplify the right-hand side of the inequality, we can rewrite it as:

2 | x - 2 | < ε.

Let δ = ε / 2, then we have:

| x - 2 | < δ => 2 | x - 2 | < 2 δ.

Substituting δ with ε / 2, we get:

2 | x - 2 | < ε / 2 * 2 = ε.

Which means that the limit of the function f(x) = 2x + 3, as x approaches 2, is indeed 7.

In summary, an Epsilon-Delta proof is a rigorous mathematical method of proving that a limit exists, by finding a suitable value of delta that corresponds to a given epsilon. It is an essential concept in calculus and real analysis that helps in understanding the convergence and continuity of functions.