Define the sequences $$ a_n = \sum_{j=1}^n \frac{1}{n} \sqrt{\frac{j}{n}} \ \text{for} \ n = 2,3,4 ... $$ $$ b_n = \sum_{j=2}^n \frac{1}{n} \sqrt{\frac{j}{n}} \ \text{for} \ n = 2,3,4 ... $$ $$ c_n = \sum_{j=1}^{n-1} \frac{1}{n} \sqrt{\frac{j}{n}} \ \text{for} \ n = 2,3,4 ... $$ In an eanrlier question I have argued for that $a_n \geq b_n$ for $n=2,3,4..$ as $$ a_n = \sum_{j=2}^n \frac{1}{n} \sqrt{\frac{j}{n}} + \frac{1}{n} \sqrt{\frac{1}{n}} = b_n + \frac{1}{n} \sqrt{\frac{1}{n}} $$ Furthermore that $b_n \geq c_n$ as $$ b_n = \sum_{j=1}^{n-1} + \frac{1}{n} + \frac{1}{n} - \frac{1}{n}\sqrt{\frac{1}{n}} $$ which means that $$ b_n + \frac{1}{n}\sqrt{\frac{1}{n}} = c_n + \frac{1}{n} $$ but as $\frac{1}{n} \geq \frac{1}{n} \sqrt{\frac{1}{n}}$for $n=2,3,4..$ this means that $b_n \geq c_n$ which I hope you can clarify as well.
I think I have to use this to find the function $f$ but I really have no idea how to start. Can you help me?