Show that the sum from 1 to n of 1/(2n+1)(2n-1) is equal to n/(2n+1) by Induction

First we check that this is true for n=1: S₁ = 1/(1x3)  which is equal to n/(2n+1) for n=1 therefore S_n = n/(2n+1) is true for n = 1. Next assume that it is true for n=k. S_k  = k/(2k+1). Now using this assumption we check that it is true for n=k+1: S_k+1 = S_k+ 1/(2(k+1) - 1)(2(k+1)+1). Rearranging this and substituting in k/(2k+1) for S_k we get S_k+1 = (k+1)/(2k+3) which is consistent with the original formula. Therefore we can say that since S_n = n/(2n+1) is true for n=1 and whenever it is true for n=k it is also true for n=k+1, it is true for all integer values of n larger than or equal to 1.