I am a physics student trying to gain a better mathematical understanding of the theory of distributions and namely the definition of the Dirac-Delta function. I understand that the defining properly of a delta function is the following integral property (integrated over the real line).
$\int f(x)δ(x-a) = f(a)$
I also understand that there are multiple delta "functions" that are the limit of so called test functions that satisfy the above integral property. My question is, for a given choice of a delta function (i.e. for a given limit of a test function), eg. $δ(x) = \lim_{a\to 0} \frac 1{|a|\sqrt{pi})} e^{-(\frac xa)^2} $ , how would you prove that this choice satisfies the above integral property and thus can be considered a valid delta function?
Thank you