A confusion in understanding the prove that the sequence ${(a_{n})^{1/n}}$ where $a_{n+m} \leq a_{n}a_{m}$ is convergent.

Question

The proof is given below:

I have a problem in understanding starting from the second paragraph. the second line in it, why liminf ${a_{n}}^{1/n}$ $\geq$ limsup ${a_{n}}^{1/n}$ will complete the proof ?

Could anyone explain this for me please?

Answer 1

We already know that $$\liminf a_n^\frac1n \le \limsup a_n^\frac1n,$$

hence, if we can show that

$$\liminf a_n^\frac1n \ge \limsup a_n^\frac1n,$$

then we have $$\liminf a_n^\frac1n = \limsup a_n^\frac1n,$$

and the limit exists.