Question

Suppose that \(\lim _{n \rightarrow \infty} t_{n}=t,\) where \(0<|t|<\infty,\) and let \(0<\rho<1\). Show that there is an integer \(N\) such that \(t_{n}>\rho t\) for \(n \geq N\) if \(t>0,\) or \(t_{n}<\rho t\) for \(n \geq N\) if \(t<0 .\) In either case, \(\left|t_{n}\right|>\rho|t|\) if \(n \geq N\).

Short answer

Question: Prove that for a convergent sequence \({t_n}\) with \(\lim_{n\to\infty}t_n = t\), there exists a positive integer N, such that for \(n\geq N\), if \(t>0,\) \(t_{n}>\rho t,\) and if \(t<0,\) \(t_{n}<\rho t\). In either case, \(\left|t_{n}\right|>\rho|t|\) if \(n \geq N\), given that \(0 < \rho < 1\). Answer: We proved that there exists a positive integer N such that for \(n \geq N\), if \(t>0,\) \(t_{n}>\rho t,\) and if \(t<0,\) \(t_{n}<\rho t\). In either case, \(\left|t_{n}\right|>\rho|t|\) if \(n \geq N\).

Step-by-step solution

Start with the definition of a limit

Recall the definition of a limit: Given a sequence \({t_n}\) and a number "t", we say that the limit of the sequence \({t_n}\) as n approaches infinity is equal to "t", if for every ε > 0, there exists a positive integer N such that for every n ≥ N, \(|t_n - t| < \varepsilon\).

Choose ε based on problem condition

Since we are given that \(0 < \rho < 1\), let's choose ε = \((1 - \rho)|t|\). Since \(|t| > 0\), ε is positive.

Determine N from the definition of limit

According to the definition of the limit, since \(\lim _{n \rightarrow \infty} t_{n}=t,\) there exists a positive integer N such that for every n ≥ N, \(|t_n - t| < \varepsilon = (1 - \rho)|t|\).

Consider the cases for t > 0 and t < 0 separately

Let's now consider the cases where t > 0 and t < 0 separately, and see whether the inequalities involving \(\rho t\) hold. Case 1: If t > 0, let's add t to the inequality \((1 - \rho)|t| > |t_n - t|\). After simplifying, we get: \([(1 - \rho + 1)|t|] > |t_n - t + t| \implies 2|t| - |t| \rho > |t_n|\). Since \(t > 0\), this simplifies to \(t_n > \rho t\), which is the required inequality. Case 2: If t < 0, let's subtract t from the inequality \((1 - \rho)|t| > |t_n - t|\). After simplifying, we get: \([(1 - \rho - 1)|t|]< -|t_n - t - t| \implies -2|t| + |t|\rho < -|t_n|\). Since \(t < 0\), this simplifies to \(t_n < \rho t\), which is the required inequality.

Show that \(|t_n| > \rho |t|\) for both cases

We have shown that \(t_n > \rho t\) when t > 0, and \(t_n < \rho t\) when t < 0. In both cases, since \(0 < \rho < 1\), we observe that \(|t_n| > \rho |t|\) for n ≥ N. Thus, we have proven that there exists a positive integer N such that for \(n \geq N\), if \(t>0,\) \(t_{n}>\rho t,\) and if \(t<0,\) \(t_{n}<\rho t\). In either case, \(\left|t_{n}\right|>\rho|t|\) if \(n \geq N\).

Key concepts

Limit of a Sequence

Understanding the limit of a sequence is foundational in real analysis. When we talk about the limit of a sequence, we are describing the behavior of the elements of the sequence as their indices grow larger and larger. Formally, we say that the sequence \(t_n\) converges to the limit \(t\) if, for every epsilon (\(\varepsilon\)) greater than zero, no matter how small, there exists a point in the sequence beyond which all the terms are within \(\varepsilon\) of \(t\).

Imagine a target centered around the value \(t\) with a radius of \(\varepsilon\). The definition tells us that after some point, all the members of the sequence will fall within this target zone. Mathematically, for every \(\varepsilon > 0\), there's an integer N such that for all \(n \geq N\), the terms of the sequence \(t_n\) satisfy the condition \(\left| t_n - t \right| < \varepsilon\). This captures the idea that as n becomes very large, \(t_n\) gets arbitrarily close to \(t\).

Thus, gaining a good understanding of the limit of a sequence is necessary for analyzing how sequences behave in the long run, which is a cornerstone concept in continuous mathematics and calculus.

Convergence

Closely related to the limit of a sequence is the concept of convergence. A sequence is said to converge if it has a limit. That is, there exists some finite number \(t\) to which the terms of the sequence get closer and closer as the sequence progresses. In our exercise, it was given that \(\lim_{n \rightarrow \infty} t_n = t\), meaning that as \(n\) gets very large, the terms \(t_n\) approach the value \(t\).

Convergence is a sign of predictability and stability in a sequence—knowing that a sequence converges gives us information about its long-term behavior. In practical terms, if you're tracking a process or phenomenon over time, and you know the associated sequence of measurements converges, then you can make future predictions with a level of confidence, assuming continuity of conditions.

Convergence is not just about getting close to \(t\), but also about staying within a desired proximity, defined by \(\varepsilon\), after a certain point. The smaller we choose \(\varepsilon\), the closer we’re forcing the terms to be to our limit \(t\). This creates an assurance that beyond a certain term \(N\), the sequence will behave well and remain within this tight range around \(t\).

Inequalities in Limits

In the realm of limits and sequences, inequalities play a crucial role. They are used to describe the behavior of sequences in relation to other numbers or sequences. Specifically, dealing with inequalities in limits allows us to set bounds on how a sequence behaves as it approaches its limit.

In our case, we had to show that for a convergent sequence \(\lim_{n \rightarrow \infty} t_n = t\), there exists a stage beyond which all terms \(t_n\) are either greater than \(\rho t\) if \(t > 0\) or less than \(\rho t\) if \(t < 0\), with \(0 < \rho < 1\). The strategy involved choosing an appropriate \(\varepsilon\) that relates to \(\rho\) and \(t\), to employ the limit definition, and then analyzing the cases for \(t\) being positive or negative separately.

Such inequalities are not just numerical acrobatics; they convey important information. In our exercise, by establishing that \(t_n > \rho t\) or \(t_n < \rho t\), we’re guaranteeing that the sequence doesn’t just get close to \(t\) but also maintains a specific relative magnitude to \(t\). This kind of assurance is vital in many applications, such as in ensuring that a sequence does not just converge, but converges at a rate or in a manner that is consistent with certain performance, stability, or safety requirements.