Nine Digits Sum

ABC+DEF=GHI. Replace each letter with a unique digit from 1 to 9, such that GHI is
  1. as small as possible.
  2. as large as possible.
Repeat for ABC+DEF=GHIJ, using the digits 0 to 9, with no leading zeros.

To avoid duplication in answers, maximize DEF in each solution. I'm interested more in analytical proofs than in computer generated solutions.

Source: Original. Based on a puzzle from The 2001 Mensa Puzzle Calendar.

Solution
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