Problem 77 - Essential Calculus

Question

Let \(\left\\{a_{n}\right\\}\) be an increasing sequence such that \(2 \leq a_{n} \leq 4\). Explain why \(\left\\{a_{n}\right\\}\) has a limit. What can you conclude about the limit?

Step-by-Step Solution

Answer

By applying the Monotone Sequence Theorem, we can conclude that the sequence \(\left\{a_{n}\right\}\) has a limit. Furthermore, the limit of this sequence is confined to the interval [2, 4].

Step 1: Recognize the Properties of the Sequence

The given sequence, \(\left\{a_{n}\right\}\), is increasing and bounded by two numbers, 2 and 4. In other words, for any n, we have \(2 \leq a_{n} \leq 4 \).

Step 2: Apply the Monotone Sequence Theorem

The Monotone Sequence Theorem states that every bounded and increasing (or decreasing) sequence has a limit. Since our sequence is increasing and bounded, we can conclude that it has a limit.

Step 3: Deduce the Possible Values of the Limit

Given the nature of the sequence (increasing and bounded), the limit of such sequence must lie in the interval [2,4]. Therefore, the limit of the sequence \(a_{n}\), if it exists, must be a number L satisfying \(2\le L \le 4\). This is derived from what's called the squeeze theorem or sandwich theorem, which states that if a sequence is squeezed between two other sequences that converge to the same limit, then that sequence also converges to that limit.