Show that the sequence $a_n = frac{sqrt n}{1+sqrt n}$ is increasing for all n.

Show that the sequence $a_n = \frac{\sqrt n}{1+\sqrt n}$ is increasing for all $n$.

To prove its increasing I need to show that $a_{n+1} \gt a_n$, however the algebra is quite tricky and I have not been successful, if anyone could give me a hint that would be greatly appreciated.

Kind Regards,

Answer

Hint:
Write it as, $$\dfrac{\sqrt{n}}{1+\sqrt{n}}=\dfrac{1+\sqrt{n}-1}{1+\sqrt{n}}=1-\dfrac{1}{1+\sqrt{n}}.$$ Since, $\sqrt{n+1}\gt\sqrt{n}$ it follows that $1+\sqrt{n+1}\gt1+\sqrt{n}$, ... can you proceed further?