Let and Evaluate
We define the independence of the integral from , for which we use the fact that the function of the form up to constant is the Fourier transform of unit step height in the interval . Consider this integral as the value at inverse Fourier transform of the product of the functions , but the contraction of the first n steps will be of compact support on the interval . Therefore, if , when calculating the convolution of this function with the last step, the width of at point , we simply calculate the area under the graph of the convolution of the first steps, but this area does not depend on .
We divide this integral into two intervals . Parameter we will suggest later, based on the conditions: . Now perform the substitution and estimate this integral on the interval :
Now we impose a condition on the parameter: a .
Now we estimate the integral on the interval :
From this representation and given the inequality , we obtain an estimate . And now we impose the condition . Let . Then the condition that means , but the condition means . Thus, when , all conditions are met.
So, on the basis of , it follows that
now let and suppose that the claim is true for then, exactly the same as above, we'll have:
so we also proved that which, with gives us
* note that