A sequence (aₙ) converges to a limit L (written lim aₙ = L) if for every ε > 0, there exists N such that n > N implies |aₙ - L| < ε. This formal definition replaces intuition: 'aₙ gets arbitrarily close to L' becomes 'aₙ stays within ε of L eventually.' It is the foundation for all rigor in calculus.
Start with 1/n → 0: given ε, find N = ⌈1/ε⌉ and verify n > N ⟹ |1/n| < ε. Then try sin(n)/n → 0 and a non-convergent sequence like (-1)ⁿ to see why the definition fails.
In calculus you learned that the sequence 1/n converges to 0 because the terms get smaller and smaller. Real analysis asks a harder question: what does "converge" actually mean, precisely enough to prove theorems with? The epsilon-N definition is the answer. It replaces the intuition "the terms get close to L" with a logical statement that can be verified or refuted by calculation.
The definition reads: (aₙ) converges to L if for every ε > 0, there exists a natural number N such that for all n > N, |aₙ − L| < ε. Parse it carefully. "For every ε > 0" is the challenge: your opponent names any positive tolerance, no matter how tiny. "There exists N" is your response: you produce a cutoff. "For all n > N, |aₙ − L| < ε" is the guarantee: from the cutoff onward, every term lies within ε of L. The logical structure ∀ε ∃N ∀n>N is the backbone of all limit proofs — N is allowed to depend on ε, and a smaller ε may require a larger N.
Here is the template applied to aₙ = 1/n and L = 0. Given any ε > 0, we want n > N to imply |1/n − 0| = 1/n < ε. Rearranging: we need n > 1/ε. Choose N = ⌈1/ε⌉ (the ceiling function gives an integer). Then for any n > N, we have n > 1/ε, so 1/n < ε. ✓ The proof has three parts: scratch work to find what N must be, a declaration of that N, and then a verification that n > N implies the desired inequality. Write it in that order every time.
A critical misconception is reading "for all n > N" as "for all n" — thinking the definition requires every term to be within ε of L. It only requires eventual closeness. Finitely many early terms can be anywhere. The sequence (1000, 2000, 1/3, 1/4, 1/5, …) converges to 0: choose N = max(2, ⌈1/ε⌉) to skip the first two large terms, and from there onward all terms satisfy 1/n < ε. Those first two terms are irrelevant to the limit because N takes care of them.
The connection to your prerequisite work on supremum and infimum is direct: when you learned to prove properties of bounds, you practiced the same logical template — given a tolerance, find a threshold that forces a quantity below it. That experience with bounding arguments is the exact skill needed to construct N given ε. As you move to the monotone convergence theorem, Cauchy sequences, and limit superior, this epsilon-N scaffolding will appear at every step. Becoming fluent with the definition now is not just about sequences — it is about internalizing the mode of reasoning that underlies all of real analysis.