Problem 28 - Advanced Calculus

Question

Show that, for any bounded sequences \(a_{n}\) and \(b_{n}\), $$ \liminf _{n \rightarrow x} a_{n}+\liminf _{n \rightarrow x} b_{n} \leq \liminf _{n \rightarrow x}\left(a_{n}+b_{n}\right) $$ and that $$ \lim _{n \rightarrow x} \sup \left(a_{n}+b_{n}\right) \leq \lim _{n \rightarrow x} \sup a_{n}+\underset{n \rightarrow x}{\lim \sup } b_{n} $$

Step-by-Step Solution

Answer

As per the definition of limit inferior and limit superior, we proved that \(\liminf_{n \rightarrow x} a_{n} + \liminf_{n \rightarrow x} b_{n} \leq \liminf_{n \rightarrow x}(a_{n} + b_{n})\) and \(\limsup_{n \rightarrow x} (a_{n} + b_{n}) \leq \limsup_{n \rightarrow x} a_{n} + \limsup_{n \rightarrow x} b_{n}\).

Step 1: Understanding Limit Inferior and Limit Superior

The limit inferior (\(\liminf\)) of a sequence is defined as the greatest lower bound of the set of its limit points, while the limit superior (\(\limsup\)) is defined as the least upper bound of the set of its limit points. In other words, the limit inferior and limit superior are the lowest and highest values that a sequence approaches without actually reaching, respectively. Understanding these definitions is crucial for solving the given exercise.

Step 2: Proving the First Inequality

Using the properties of limit inferior (\(\liminf\)), we know that for any \(x\), \(\liminf_{n \rightarrow x} a_{n}\) and \(\liminf_{n \rightarrow x} b_{n}\) are less than or equal to \(a_{n}\) and \(b_{n}\) respectively. Adding these two inequalities we have: \(\liminf_{n \rightarrow x} a_{n} + \liminf_{n \rightarrow x} b_{n} \leq a_{n} + b_{n}\). Since this inequality is valid for every \(x\), taking the \(\liminf\) on both sides gives us \(\liminf_{n \rightarrow x} a_{n} + \liminf_{n \rightarrow x} b_{n} \leq \liminf_{n \rightarrow x}(a_{n} + b_{n})\), proving the first inequality.

Step 3: Proving the Second Inequality

Using the properties of limit superior (\(\limsup\)), we know that for all \(x\), \(a_{n} + b_{n}\) are less than or equal to \(\limsup_{n \rightarrow x} a_{n}\) and \(\limsup_{n \rightarrow x} b_{n}\). Hence it follows that \(\limsup_{n \rightarrow x} (a_{n} + b_{n}) \leq \limsup_{n \rightarrow x} a_{n} + \limsup_{n \rightarrow x} b_{n}\), proving the second inequality.

Key Concepts

Sequences and Series

When diving into the world of sequences and series, it's important to first understand what a sequence is. A sequence is essentially a list of numbers that follows a specific order governed by a rule or a formula. They can be finite or infinite and are indexed by natural numbers, often denoted as \(a_n\) or \(b_n\) where \(n\) is your index number.

Series, on the other hand, involves adding these sequences together. An infinite series is the sum of all terms in an infinite sequence. Understanding both these concepts is crucial when dealing with limits, as they form the backbone of many theorem applications in calculus.

For sequences that are bounded, this means they don't extend beyond certain fixed values, known as the lower and upper bounds. This boundedness is a vital property when determining limit superior and limit inferior, especially in proving inequalities like those in our exercise.

Inequalities

Inequalities are mathematical expressions that involve the symbols \(<, >, \leq, \geq\). They show the relationship between two expressions. In our exercise, inequalities help us to understand the behavior of the limit inferior and superior in bounded sequences.

When you compare two numbers using inequalities, you're establishing which one is larger or smaller. For instance, the inequality \(a \leq b\) implies that \(a\) is either less than or equal to \(b\).

In the provided exercise, inequalities are fundamental to demonstrating that the sum of the limit inferiors of two sequences is less than or equal to the limit inferior of their sum, and similarly for their limit superiors. These inequalities stem from the definitions of \(\liminf\) and \(\limsup\), serving as essential tools in demonstrating these relationships under the context of calculus.

Calculus

Calculus is the branch of mathematics that deals with the change of functions and the accumulation of quantities. It's foundational for understanding the behavior of sequences and their limits, forming the theoretical backbone of exercises like the one given.

One of the core components of calculus is the concept of limits which describe how a sequence behaves as it approaches a certain value. Limit inferior and limit superior are important variations of limits that provide more specific information about a sequence's behavior.

The limit inferior, \(\liminf\), represents the greatest value that a bounded sequence doesn't permanently exceed. Conversely, the limit superior, \(\limsup\), is the smallest value that a sequence doesn't permanently drop below. In context, these concepts are used within inequalities to show how combined sequences behave in relation to their component sequences, specifically emphasizing the boundaries as they approach those ultimate values.