Cauchy sequences and completeness
General · Mathematics
Study notes
Q: Show 1/n is Cauchy. Why does Cauchy ⟺ convergent need completeness? |1/m - 1/n| ≤ 1/m + 1/n < ε for m,n > 2/ε ✓ CAUCHY! Convergent → Cauchy: |aₘ-aₙ| ≤ |aₘ-L|+|aₙ-L| < ε! Cauchy → convergent: needs LIMIT to exist - ℝ complete ✓, ℚ ✗! (√2 approximations: Cauchy in ℚ, limit missing!)