Question

How do we know that lim sn is always meaningful for monotone sequences?

A. Monotone sequences are always bounded, ensuring a well-defined limit.

B. Monotone sequences are always decreasing, guaranteeing convergence.

C. Monotone sequences are either always increasing or decreasing, allowing for the existence of limits.

D. Monotone sequences always have a finite number of terms, ensuring convergence.

Answer 1

Monotone sequences can have a well-defined limit because they are always bounded, either increasing or decreasing, and are convergent.

Step-by-step explanation:

We know that lim sn is always meaningful for monotone sequences because:

1. Monotone sequences are always bounded, ensuring a well-defined limit. This means that the sequence does not go to infinity or negative infinity.
2. Monotone sequences are always either increasing or decreasing, allowing for the existence of limits. This means that the sequence is moving towards a specific value.
3. Monotone sequences are always convergent because they are either always increasing and bounded above or always decreasing and bounded below. This guarantees that the sequence will approach a specific value.