Question

Prove Theorem 2 , with (i) replaced by the weaker assumption ("subuniform limit") $$ (\forall \varepsilon>0)(\exists \delta>0)\left(\forall x \in G_{\neg p}(\delta)\right)\left(\forall y \in G_{\neg q}(\delta)\right) \quad \rho(g(x), f(x, y))<\varepsilon $$ and with iterated limits defined by $$ s=\lim _{x \rightarrow p} \lim _{y \rightarrow q} f(x, y) $$ iff \((\forall \varepsilon>0)\) $$ \left(\exists \delta^{\prime}>0\right)\left(\forall x \in G_{\neg p}\left(\delta^{\prime}\right)\right)\left(\exists \delta_{x}^{\prime \prime}>0\right)\left(\forall y \in G_{\neg q}\left(\delta_{x}^{\prime \prime}\right)\right) \quad \rho(f(x, y), s)<\varepsilon $$

Short answer

The theorem is proven by showing subuniform limits guarantee iterated limits.

Step-by-step solution

Understanding Given Assumptions

We start by interpreting the given assumptions. The subuniform limit assumption implies that for any small positive number \(\varepsilon\), there is another positive number \(\delta\) such that for any \(x\) and \(y\) near the points where propositions \(p\) and \(q\) fail (denoted as \(G_{eg p}(\delta)\) and \(G_{eg q}(\delta)\)), the distance between the images under a function \(g\) and \(f\), \(\rho(g(x), f(x, y))\), is smaller than \(\varepsilon\). This sets a constraint on how \(g(x)\) approximates \(f(x, y)\) when \(x\) and \(y\) approach these limit points.

Examining Iterated Limits Definition

The iterated limits definition provides a way to express the limit \(s = \lim_{x \rightarrow p} \lim_{y \rightarrow q} f(x, y)\) using nested \(\varepsilon-\delta\) descriptions. This means for any small \(\varepsilon > 0\), there exists a \(\delta' > 0\) such that for any \(x\) close to \(p\), specifically in \(G_{eg p}(\delta')\), we can find another \(\delta_x'' > 0\) ensuring that for all \(y\) close to \(q\), in \(G_{eg q}(\delta_x'')\), the distance \(\rho(f(x, y), s)\) is less than \(\varepsilon\). This defines \(s\) as the limit value.

Establishing Equality of Limits

We need to show how the given subuniform limit assumption can lead to the specification of the iterated limits. Assume the subuniform condition holds: for \(\varepsilon > 0\), there is \(\delta > 0\) such that \(\rho(g(x), f(x, y)) < \varepsilon/2\). By leveraging similar logic for iterated limits along with the convergence \(\rho(g(x), s) < \varepsilon/2\) (implied by the limits \(x \rightarrow p\)), the triangle inequality implies \(\rho(f(x, y), s) < \varepsilon\). This proves that \(s\) is the iterated limit by meeting the criteria of both limits as \(x \rightarrow p\) and \(y \rightarrow q\).

Conclusion of Proof

The theorem requires showing how the conclusion derived from subuniform limits fits within the iterated limits framework. Since the sequence of logical deductions ensures that every \(\varepsilon > 0\) condition for the subuniform limit directly leads to an \(\varepsilon\) condition in the iterated framework, the substitution leads to demonstration of the theorem. The set conditions ensure the limit \(s\) holds under both definitions, proving the theorem.

Key concepts

Theorem Proving

Theorem proving in mathematical analysis is a structured process used to demonstrate the validity of a given statement based on logical reasoning and previously established results. The essence of theorem proving lies in detailed steps that connect assumptions with a conclusion.
This involves understanding initial conditions, employing definitions, and using logical constructs to arrive at a valid and supported conclusion. A critical part of proving theorems is using previously established axioms and other known theorems, which can lend credibility to new results.
The process typically starts with analyzing given assumptions, building a framework of logical steps, and using these steps to derive the final statement or conclusion that needs to be proven.

Limit Theorems

Limit theorems provide the foundation for understanding the behavior of functions as inputs approach a particular value. They help describe the behavior of functions near specific points, which is crucial for calculus and analysis.
In our context, the discussion is centered around limits involving two variables. The limit theorems for such cases often require examining the function values as both variables approach their respective limit points. This is especially important when handling functions of more than one variable, like in the subuniform and iterated limits.
These theorems help in establishing whether a function behaves continuously or if it approaches a particular limit value under given constraints. They form the backbone of continuity and differentiability concepts critical in advanced mathematics.

Epsilon-Delta Definition

The epsilon-delta definition is a precise mathematical way to express limits. It's a language used to define how closely a function gets to a given limit under certain conditions. The approach establishes a relationship between two small positive numbers, \( \varepsilon \) (epsilon) and \( \delta \) (delta).

• Epsilon (\( \varepsilon \)): A small positive number that represents how close the function's output needs to be to the limit value.
• Delta (\( \delta \)): Another small positive number denoting how near the input value should be to the target value in order for the function's output to stay within \( \varepsilon \) of the limit.

By using \( \varepsilon \) and \( \delta \), mathematicians provide a rigorous framework to ensure that the function values remain close to the limit for inputs sufficiently near a specified point.
This technique is fundamental for proving continuity and convergence properties within mathematical analysis.

Iterated Limits

Iterated limits involve evaluating the limit of a function of two or more variables sequentially, one variable at a time. This happens by fixing one variable and finding the limit with respect to another, and then repeating the process for the remaining variable.
This concept is useful for scenarios where it's challenging to evaluate a joint limit directly. By analyzing iterated limits, one can understand the step-by-step behavior of a multivariable function as each input approaches its respective limit.
It is essential to note that the iterated limits approach may not always yield the same result as the direct joint limit due to potential dependencies between the variables. The technique is instrumental in expanding our understanding of multivariable functions and their limits.