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Exercise 6.1.6 (Formal limits are genuine limits)

Prove Proposition 6.1.15, using the following outline.

Proposition 6.1.15. Suppose that (an)n=1∞ is a Cauchy sequence of rational numbers. Then (an)n=1∞ converges to LIMn→∞an, i.e.

LIMn→∞an = lim ⁡ n→∞an.