Definition of Limit

By the definition of limit, we have:

For a sufficiently small positive values ε1 and ε2, we can say that:

When |x - c| < ε1, |f(x) - L| < δ

AND

When |x - c| < ε2, |g(x) - M| < δ

where δ is also a sufficiently small positive value.

Now, for f(x) - g(x), we can find ε3 < ε1 and ε4 < ε2 such that:

When |x - c| < ε3, |f(x) - L| < δ/2

AND

When |x - c| < ε4, |g(x) - M| < δ/2

We now have, when |x - c| < min(ε3, ε4):

|[f(x) - L] - [g(x) - M]| <= |f(x) - L| + |g(x) - M|

|[f(x) - g(x)] - (L - M)| <= δ

Hence lim (x → c) [f(x) - g(x)] = L - M

2009-12-10 20:58:19 補充：
沒問題, 因為 ε1, ε2, ε3 和 ε4 都只說明是 very small, 並沒有一個 definite 的 value.