Statement#

Prove that the sequence $\{a_n\}$ defined by $a_n = \frac{1}{n}$ converges to $0$ as $n \to \infty$.

Use the $\varepsilon$-$\delta$ definition of sequence convergence to construct a rigorous proof.

Required Topics#

• Definition of sequence convergence
• Epsilon-delta proofs
• Archimedean property of real numbers
• Inequality manipulation

Definition of Convergence#

A sequence $\{a_n\}$ converges to $L$ if:

For every $\varepsilon > 0$, there exists $N \in \mathbb{N}$ such that for all $n > N$: $|a_n - L| < \varepsilon$

What to Prove#

Show that $\lim_{n \to \infty} \frac{1}{n} = 0$ by finding an explicit $N$ in terms of $\varepsilon$.

Solution#

Solution coming soon.