Continuing Exercises 5 and 4, to show that the inner product (6.14) is well defined, a number of facts have to be checked.

(a) Verify that the limit converges.

(b) Verify that the limit is independent of the Cauchy sequences used.

(c) Verify that the properties of an inner product are satisfied. The only non-trivial property to verify is that \(\langle

f, fangle_{\mathscr{G}}=0\) if and only if \(f=0\).