The prob of 2 successes out of 2n is
n(2n-1)p^2(1-p)^{2n-2}
the prob of 1 out of n is
np(1-p)
to show the ratio is less than 1, you only need to show
p(1-p)^{n-1}<1/(2n-1)
But p(1-p)^{n-1} is max'd at p=1/n with a value of (n-1)^{n-1}/n^n
Now it's up to you to show the inequality.
