Question

Prove: The sequence \(\left\\{s_{n}\right\\}\) is bounded if, for some positive \(\epsilon,\) there is an integer \(N\) such that \(\left|s_{n}-s_{m}\right|<\epsilon\) whenever \(n, m \geq N\)

Short answer

Question: Prove that a sequence \(\left\{s_{n}\right\}\) is bounded if there exists a positive \(\epsilon\) and an integer \(N\) such that \(\left|s_{n}-s_{m}\right|<\epsilon\) whenever \(n, m \geq N\). Answer: To prove that the sequence \(\left\{s_{n}\right\}\) is bounded, we found constants \(L\) and \(M\) such that \(L \leq s_{n} \leq M\) for all \(n \in \mathbb{N}\). We first identified the maximum and minimum values of \(s_n\) for \(n\) less than \(N\) and then created upper and lower bounds for \(s_n\) for \(n \geq N\). Finally, we combined the derived upper and lower bounds to show that the sequence is indeed bounded with \(L \leq s_n \leq M\) for all \(n \in \mathbb{N}\).

Step-by-step solution

Identify the maximum value of \(s_n\) for \(n\) less than \(N\)

Choose the maximum value of \(s_n\) for \(n\) less than \(N\). Suppose the maximum value is reached by \(s_p\) with \(p < N\). Let \(M_1 = \max\{s_1,s_2,\dots,s_{N-1}\}\) and \(M_1 = s_p\). We will later use \(M_1\) to create the upper bound for \(n < N\).

Create an upper bound for \(s_n\) for \(n \geq N\)

We know that \(\left|s_{n}-s_{m}\right|<\epsilon\) whenever \(n, m \geq N\). So, when \(n = N\) and \(m \geq N\), $$\left|s_{m} - s_{N}\right| < \epsilon$$ Since \(s_m \geq s_N - \epsilon\), we have $$s_m \leq s_N + \epsilon$$ Let \(M_2 = s_N + \epsilon\) as the upper bound for \(s_m\) for \(m \geq N\).

Create a single upper bound for the sequence

We want a single value \(M\) to be an upper bound for the sequence. For all \(n \in \mathbb{N}\), if \(n < N\), then \(M_1\) is the upper bound, and if \(n \geq N\), then \(M_2\) is the upper bound. Thus, let \(M = \max\{M_1, M_2\}\). So, \(s_n \leq M\) for all \(n \in \mathbb{N}\).

Identify the minimum value of \(s_n\) for \(n\) less than \(N\)

Now, we find the minimum value of \(s_n\) for \(n\) less than \(N\). Suppose the minimum value is reached by \(s_q\) with \(q < N\). Let \(L_1 = \min\{s_1,s_2,\dots,s_{N-1}\}\) and \(L_1 = s_q\). We will later use \(L_1\) to create the lower bound for \(n < N\).

Create a lower bound for \(s_n\) for \(n \geq N\)

We already know that \(\left|s_{n}-s_{m}\right|<\epsilon\) whenever \(n, m \geq N\). So, when \(n = N\) and \(m \geq N\), $$\left|s_{m} - s_{N}\right| < \epsilon$$ Since \(s_m \leq s_N + \epsilon\), we have $$s_m \geq s_N - \epsilon$$ Let \(L_2 = s_N - \epsilon\) as the lower bound for \(s_m\) for \(m \geq N\).

Create a single lower bound for the sequence

We want a single value \(L\) to be a lower bound for the sequence. For all \(n \in \mathbb{N}\), if \(n < N\), then \(L_1\) is the lower bound, and if \(n \geq N\), then \(L_2\) is the lower bound. Thus, let \(L = \min\{L_1, L_2\}\). So, \(s_n \geq L\) for all \(n \in \mathbb{N}\).

Conclude the sequence is bounded

Combining the derived upper and lower bounds, we see that \(L \leq s_n \leq M\) for all \(n \in \mathbb{N}\). Therefore, the sequence \(\left\{s_{n}\right\}\) is bounded.

Key concepts

Real Analysis

Real analysis is a substantial branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. It concerns the rigorous study of limits, sequences, continuity, differentiation, and integration. For sequences in particular, real analysis seeks to understand the behavior of the elements within the sequence as they progress towards infinity.

Central to this study is the concept of 'sequence convergence' and the conditions under which a sequence is 'bounded'. To grasp the depth of these concepts, we delve into the realm of sequences—a list of numbers in a specific order based on a certain rule. The focus on understanding the attributes of these sequences, such as bounds and convergence, helps to build the foundational knowledge needed for more complex calculus and higher mathematics.

Sequence Convergence

Within real analysis, sequence convergence is the process of approaching a specific value, or limit, as the sequence's index increases indefinitely. In simpler terms, as you move further along the list of numbers in a sequence, they get closer and closer to a certain number called the limit. A sequence that doesn't converge is said to be divergent. It is important to note that not all sequences have a limit; those that do are called 'convergent sequences'.

The formal definition uses the notion of 'arbitrarily close', which means for any small distance \( \epsilon > 0 \), there exists a point in the sequence after which all the sequence's terms are within that distance from the limit. Such intricate definitions can sometimes overwhelm students, but understanding convergence is crucial for appreciating the broader strokes of calculus and the behavior of functions.

Upper and Lower Bounds

To say that a sequence is bounded means that there is a real number, or a 'bound', that the sequence doesn't exceed. Specifically, a sequence \( \left\{s_n\right\} \) is bounded above if there exists a number M such that \( s_n \leq M \) for all n. This M is called an 'upper bound' of the sequence. Similarly, a sequence is bounded below if there exists a number L such that \( s_n \geq L \) for all n, with L being a 'lower bound'.

When a sequence has both an upper and lower bound, it's simply 'bounded'. This does not necessarily say anything about convergence but rather about containment within a specific numerical region. Bound sequences are important because they ensure that the sequence won't 'explode' into infinity in either direction, which implies a certain level of predictability and stability within the sequence.