We have the following sequence a₁=1 , a_n+1= a_n+ 1/a_n and so forth for n=1,2,…


Show that ln n < a_n < n for all n>1.

What would be the best approach for this kind of problem?

The right hand side of the inequality is trivial. For the left hand side consider that the definition of ln(n) is the area under the y=1/x curve, starting at x=1 out to x=n. Compare that to an approximation of the area under y=1/x using the sum of areas of rectangles.