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Understanding the concept of the limit of a sequence is fundamental for analyzing the behavior of sequences as they progress towards infinity. In simple terms, the limit of a sequence is the value that the sequence's terms approach as the index (often denoted as 'n') goes to infinity. This concept tells us about the long-term trend of the sequence, whether it settles down to a single value, oscillates between various values, or grows without bound.

When we say that a sequence \( a^n \) converges to 0 for -1 < 'a' < 1 as n approaches infinity, we're describing the sequence's end behavior. As we choose larger and larger values for 'n', the values of \( a^n \) get closer to 0. This convergent trend is highly dependent on the characteristics of the number 'a', specifically that it lies between -1 and 1 but not including the endpoints.

The absolute value of a number can be thought of as the number's distance from zero on the number line, regardless of direction. Symbolically, the absolute value of 'a' is written as \( |a| \). In our exercise, considering the absolute value is a clever simplification because it lets us focus on the magnitude of 'a' without worrying about its sign.

For our sequence, since -1 < 'a' < 1, taking the absolute value means we are dealing with numbers in the range of 0 < \( |a| \) < 1. This observation is pivotal because it tells us that \( |a| \) is a proper fraction, and proper fractions have properties that are helpful when raised to increasing powers—their values decrease, approaching zero.

Exponents indicate the number of times a base number is multiplied by itself. A fundamental property of exponents that we utilize in this exercise is that when you raise a number between 0 and 1 (a proper fraction) to a positive integer power, the result is smaller than the base number. More formally, if 0 < 'a' < 1 and 'n' is an integer greater than 1, then \( a^n < a \).

The consequence of this property is that as 'n' becomes larger, the value of \( a^n \) becomes progressively smaller. Hence, for our sequence \( a^n \) with \( -1 < a < 1 \), each additional exponentiation means the term \( a^n \) is a fraction of its prior size, driving the value of the sequence closer to 0.

A rigorous mathematical definition of limit for sequences was established to precisely capture the intuitive idea of convergence. According to this definition, a sequence \( a^n \) has a limit L if, for every positive number epsilon (ε), no matter how small, there exists a positive integer N such that for all 'n' greater than N, the absolute value of \( a^n - L \) is less than ε.

In the context of our problem, this definition translates to the following: we can make the terms of our sequence \( a^n \) as close to 0 as we like (within any desired margin of ε) by choosing a large enough 'n'. Thus, when we prove that, given \( 0 < |a| < 1 \), there exists such an 'N' for any ε, we affirm that the limit of \( a^n \) as n approaches infinity is indeed 0.