Of course, my solution is not the most elegant, but I think it is quite reasonable.
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