To prove that:

$\lim_{n \to \infty} \left( 1 - \frac{1}{n^2} \right) = 1$

using the Sandwich Theorem (also known as the Squeeze Theorem), we need to find two sequences that bound the given expression, both of which converge to 1 as $n \to \infty$.

Step-by-Step Proof:

1. Expression Setup: We are interested in the limit: $\lim_{n \to \infty} \left( 1 - \frac{1}{n^2} \right)$

2. Bounding the Function: Notice that for $n \geq 1$, $\frac{1}{n^2} \geq 0$. This gives the inequality: $0 \leq \frac{1}{n^2} \leq 1$ Subtracting $\frac{1}{n^2}$ from 1, we get: $1 - \frac{1}{n^2} \leq 1$ At the same time, since $\frac{1}{n^2} \geq 0$, we also have: $1 - \frac{1}{n^2} \geq 1 - 0 = 1$

3. Squeeze the Function: Combining both inequalities, we get: $1 \leq 1 - \frac{1}{n^2} \leq 1$

4. Applying the Sandwich Theorem: Since both bounds are equal to 1 and the expression $1 - \frac{1}{n^2}$ is squeezed between them, by the Sandwich Theorem, we can conclude: $\lim_{n \to \infty} \left( 1 - \frac{1}{n^2} \right) = 1$

Thus, the limit is proven to be 1 as $n \to \infty$.

Would you like further details or have any questions? Here are five related questions that might help you understand this concept better:

1. How does the Sandwich Theorem work in general, and why is it applicable here?
2. Could we use any other method (like direct evaluation) to solve this limit?
3. What happens if we change the function to $1 - \frac{1}{n^k}$ for any $k > 2$?
4. How would this proof change if the function was $1 + \frac{1}{n^2}$ instead of $1 - \frac{1}{n^2}$?
5. Can the Sandwich Theorem be used to prove limits involving more complex functions, like trigonometric functions?

Tip: When applying the Sandwich Theorem, always ensure that the function you're working with is properly bounded by two other functions that converge to the same limit!