real-analysis-fundamentals

Epsilon-N proofs formalize sequence convergence by showing that for any small tolerance (epsilon), you can find a point (N) beyond which all terms stay within that tolerance of the limit. Start by assuming epsilon > 0, work backward from the inequality |aₙ - L| < epsilon to find N, then verify the logical chain forward.

Pointwise continuity fixes a point first, then finds delta for each epsilon; uniform continuity finds one delta that works for all points simultaneously. Uniform continuity is stronger and guarantees continuous functions on closed intervals map to compact sets, enabling powerful theorems like preservation of compactness.

The Ratio Test examines the limit of consecutive term ratios: compute L = lim|aₙ₊₁/aₙ|. If L < 1, the series converges absolutely; if L > 1, it diverges; if L = 1, the test is inconclusive. This method handles exponentials and factorials efficiently.

A Cauchy sequence has terms that eventually cluster arbitrarily close together: for any epsilon > 0, terms beyond some N satisfy |aₘ - aₙ| < epsilon. In complete spaces like ℝ, every Cauchy sequence converges, providing an alternative characterization of convergence without knowing the limit beforehand.

Real analysis develops continuity rigorously on ℝ using epsilon-delta; these foundations generalize to metric spaces and topological spaces through open sets. Understanding limits and sequences in ℝ builds intuition for abstract continuity but requires additional framework for non-Euclidean contexts.

The Root Test (examining lim ⁿ√|aₙ|) is powerful for series with nth powers but fails when the limit equals 1, just like the Ratio Test. It's less practical for factorials and alternating series; Ratio and specialized tests (Alternating, Integral) often provide clearer results in those cases.