91Ó°ÊÓ

**Question:** Prove using the formal definition of the limit of a sequence that: $$\lim_{n \rightarrow \infty} \frac{1}{n} = 0$$ **Answer:** We can prove that $$\lim_{n \rightarrow \infty} \frac{1}{n} = 0$$ by the following steps: 1. Write the definition of the limit of a sequence: $$\lim_{n \rightarrow \infty} a_n = L$$ if for any real number \(\epsilon > 0\), there exists a positive integer \(N\) such that for all \(n > N\), we have $$\left| a_n - L \right| < \epsilon$$ 2. Apply the definition to the given limit: In our case, \(a_n = \frac{1}{n}\) and \(L = 0\). We need to find \(N\) such that for all \(n > N\), $$\left| \frac{1}{n} - 0 \right| < \epsilon$$ 3. Simplify the inequality: We can rewrite the inequality as $$\frac{1}{n} < \epsilon$$ 4. Find the value of \(N\): By solving the inequality for \(n\), we get $$n > \frac{1}{\epsilon}$$. We choose the smallest integer \(N\) greater than \(\frac{1}{\epsilon}\), that is, $$N = \lceil \frac{1}{\epsilon} \rceil$$ 5. Conclude the proof: Since we've found an integer \(N\) such that for all \(n > N\), \(\frac{1}{n} < \epsilon\), we have proven that $$\lim_{n \rightarrow \infty} \frac{1}{n} = 0$$ by the formal definition of the limit of a sequence.