I want to calculate the limit of the sequence where $a_n = \bigl(1 - \frac{1}{n}\bigr) * \bigl(1 - \frac{2}{n}\bigr) * \dots * \bigl(1 - \frac{n-1}{n}\bigr)$ as $n \to \infty$. I want to use compactness theorem to prove the result.
Compactness Theorem : If any statement is true on any finite sub-setof sentences $\Gamma$ then the statement is true on the whole set ofsentences $\Gamma$.(Where $\Gamma$ is usually a countable infinite set ofsentences)
Now the limit of the sequence formed by finite number of terms is always 1. For example when only taking the terms $\bigl(1 - \frac{1}{n}\bigr),\bigl(1 - \frac{4}{n}\bigr), \bigl(1 - \frac{n - 5}{n}\bigr)$. That is the limit of the sequence $$\lim_{n \to \infty}\bigl(1 - \frac{1}{n}\bigr)*\bigl(1 - \frac{4}{n}\bigr)*\bigl(1 - \frac{n-5}{n}\bigr) = 1$$.
Then can I claim that $$\lim_{n \to \infty}\prod_{i = 1}^{i = n -1 }\bigl(1 - \frac{i}{n} \bigr) = 1$$