Someone told me this wasn't correct?

Could anyone clarify please

I wanted to show that

two ways

Method one

Since

and also since as n gets sufficiently large there

So from there I set up my inequalities

and

Knowing the left and right integrals we see that

by the squeeze theorem

Method two

Since as

and

and just to completely be thorough I included this step

could somone point me in the direction of my mistake? I am sure it is a wrong assumption

Thanks...Mathstud