gen-define-the-limit-of-a-sequence-and-give-one-short-worked-exa-6c41

Define the limit of a sequence and give one short worked example.

# Limit of a Sequence ## Definition A sequence $(a_n)$ has **limit** $L$ as $n \to \infty$ if: For every $\epsilon > 0$, there exists a positive integer $N$ such that whenever $n > N$, we have $|a_n - L| < \epsilon$. We write: $\lim_{n \to \infty} a_n = L$ or $a_n \to L$ as $n \to \infty$. **Informally:** The terms of the sequence get arbitrarily close to $L$ and stay close as $n$ gets large. --- ## Worked Example **Prove that** $\lim_{n \to \infty} \frac{1}{n} = 0$ **Proof:** Let $\epsilon > 0$ be given. We need to find $N$ such that when $n > N$: $$\left|\frac{1}{n} - 0\right| < \epsilon$$ This simplifies to: $$\frac{1}{n} < \epsilon$$ $$n > \frac{1}{\epsilon}$$ **Choose** $N = \lceil \frac{1}{\epsilon} \rceil$ (the ceiling of $1/\epsilon$). Then for all $n > N$: $$n > \frac{1}{\epsilon} \implies \frac{1}{n} < \epsilon \implies \left|\frac{1}{n} - 0\right| < \epsilon$$ Therefore, $\lim_{n \to \infty} \frac{1}{n} = 0$. ∎