Okay. The left side is clearly <a,b>. The right side is...um?
Let j = (1,2,...,n) and k = (1,1/2,...,1/n), then <sqrt(j),a><sqrt(k),b>.
I guess? I don't have any idea what to do with this.
Since the indices are the same the scalar j can cancel out as every term of the first sum will have (1/j) and every term of the second will have a j. It is simply the Cauchy-Schwarz inequality now.
Cauchy?Schwarz inequality - Wikipedia, the free encyclopedia