|Proof of scaling property of impulse function|
Well I messed around with it for a bit and found one way to do it- take the definition of the delta function before the limit- like this:
0 for x < -a
delta(a, x) = 1/2a for -a < x < a
0 for x > a
and substitude k(t-to) for x...you'll find after messing around with it that a gets pulled in by a factor of 1/k. The area becomes 1/k and but the function is not translated, and hence your property is proved, although how rigorous this would be considered by mathematical standards I don't know.