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Understanding the sequence \( \left\{ \frac{n}{n+1} \right\} \) is crucial for analyzing its behavior as \( n \) increases. This sequence is defined as each term being \( \frac{n}{n+1} \), where \( n \) is a natural number. A natural number is a positive integer starting from 1, so examples of such numbers include 1, 2, 3, and so on. By reworking the expression, each term can be seen as \( \frac{n}{n+1} = 1 - \frac{1}{n+1} \). This is a significant simplification because it shows how the term evolves: as \( n \) increases, \( \frac{1}{n+1} \) diminishes, meaning the sequence moves closer and closer to 1.
This simplification helps us understand the sequence's behavior without computing numerous terms. Knowing the expression \( 1 - \frac{1}{n+1} \) highlights its tendency to approach a particular value, forming the foundation for further analysis in terms of bounds and convergence.

Convergence is all about determining where a sequence is headed as its terms progress. With the sequence \( \left\{ \frac{n}{n+1} \right\} \), convergence implies finding the value the sequence terms approach as \( n \) becomes very large.
By observing the expression \( 1 - \frac{1}{n+1} \), we note that \( \frac{1}{n+1} \) shrinks towards zero as \( n \) grows larger. Consequently, each term of the sequence approaches 1. In formal terms, we say that the sequence \( \left\{ \frac{n}{n+1} \right\} \) converges to 1.

• This outcome is significant because convergence informs us about the behavior of the entire sequence.
• It helps analyze whether a sequence reaches a certain limit.

Thus, the sequence illustrates the pattern of diminishing differences from 1, highlighting the least upper bound concept.

Inequalities play a vital role in proving the least upper bound of a sequence. For our sequence \( \left\{ \frac{n}{n+1} \right\} \), the concept of inequalities comes into play when we need to show that for any number \( M < 1 \), we can find a term in the sequence larger than \( M \).
Let's break it down: given that \( \frac{n}{n+1} = 1 - \frac{1}{n+1} \), showing that each term exceeds \( M \) when \( M < 1 \) can be translated to finding an \( N \) such that \( \frac{n}{n+1} > M \) for all \( n > N \).

• This inequality is rearranged to state \( \frac{1}{n+1} < 1 - M \), guiding us to solve for \( N \) as \( n > \frac{1}{1-M} - 1 \).
• Setting \( N = \left\lceil \frac{1}{1-M} - 1 \right\rceil \), guarantees that for any \( n > N \), the terms satisfy \( \frac{n}{n+1} > M \).
• This ensures that 1 remains the sequence’s least upper bound, as each term eventually surpasses any \( M < 1 \).

Mastering inequalities is essential in establishing bounds for sequences, validating that values beyond a certain point exceed predefined limits.