how to prove φ(n) tends to infinity as $n$ grows?

Question

I am wondering how can I prove $\lim\limits_{n \rightarrow \infty} {φ(n)=\infty}$

My attempt: For a prime number, we have φ(n)=n-1, so the equation above is proved. However, how can I prove it when n is a composite number?

What do you think about it? Could you please show me?

Regards

Answer 1

Since we have the following inequality: $$\varphi(n) \ge \sqrt{\frac{n}{2}}$$ it is obvious that $\lim_{n\to\infty}\varphi(n)=\infty$ holds.