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In the realm of real analysis, limits help us understand what happens to a function or sequence as we approach a particular point. Talking about limits of sequences, we're interested in what values the sequence approaches as the index grows infinitely large.

The notation \(\lim_{n \to \infty} a_n \) refers to the limit of the sequence \(a_n\) as \(n\) approaches infinity. If this limit exists and is a specific number \(L\), it means that for any arbitrarily small positive number \(\epsilon\), there exists a point after which all the sequence terms \(|a_n - L|\) are less than \(\epsilon\).

For the sequence \(n^2 / n!\), we analyze how it behaves as \(n\) increases. Through the limit definition, we validate that \(n^2/n!\) converges to zero.

A sequence is an ordered list of numbers generated by some mathematical rule. In real analysis, sequences are fundamental in understanding functions, continuity, and limits.

Consider the sequence \(n^2 / n!\):

• \(n^2\) grows rapidly because it's a power of \(n\).
• \(n!\) (n factorial) grows even quicker because it's the product of all positive integers up to \(n\).

As \(n\) increases, \(n!\) outpaces \(n^2\), causing the fraction \(n^2 / n!\) to shrink. This behavior helps to intuitively grasp why the sequence approaches zero.

The Squeeze Theorem, or Sandwich Theorem, is a crucial tool when evaluating limits. It allows us to determine the limit of a sequence by comparing it to two other sequences whose limits are known. If a sequence \(a_n\) is 'squeezed' between two sequences \(b_n\) and \(c_n\), and both \(\lim_{n \to \infty} b_n = L\) and \(\lim_{n \to \infty} c_n = L\), then \(\lim_{n \to \infty} a_n = L\).

For \(n^2/n!\), we found that it is less than or equal to \(1/n\) for \(n \geq 2\). Since \(1/n\) approaches zero as \(n\) goes to infinity, we apply the Squeeze Theorem to conclude that \(n^2/n!\) must also approach zero.

Using the Squeeze Theorem simplifies the task of finding limits, especially when direct computation is challenging.