Looks OK (except that your value isn't an integer, but that's pretty easy to fix). Here's a similarly short proof (denoting nE = \{nx | x ∈ E\}):

The set \{x ∈ ℕ | x is an upper bound for nE \} is non-empty, so has a least element by the well-ordering theorem (if you prefer not to assume that, \{x ∈ 1, ..., K | x is an upper bound for nE\} is finite, so clearly has a smallest element). That smallest element (call it y) is still an upper bound for nE, so y/n is an upper bound for E. But y - 1 is not an upper bound for nE, so (y - 1)/n is not an upper bound for E.