We are required to use the sandwich/squeeze theorem to find the following limit : $$\lim_{n \hspace{0.1cm}\to \hspace{0.1cm} \infty} n^{1/n} \hspace{0.5cm}\forall \hspace{0.3cm} n \in \mathbb{N}$$
The sequence that is lesser than the above sequence can be easily identified as $1^{1/n}$
I am stuck with the sequence to be found for the right part of the inequality.
I saw in a Youtube video that this same question was asked but they had provided in the question itself that the student may use the following inequality without proof :
$$n^{1/n} < 1 + 2\sqrt{\frac{1}{n}}$$
It is quite easy to prove after this that the the given sequence converges to $1$ as n gets larger and larger.
- I am interested in proving the given equality
- I am looking for any other sequence that could be used to fill in the inequality
Till now what I have done for problems like this is as follows. I require a greater sequence than $n^{1/n}$. So what I think immediately is that I must either increase the base n or decrease the n in the power so overall the sequence will be greater.
Increasing the base from n to $n^n$ doesn't help here. Decreasing the power n to 1 doesn't help either since in both cases we get n as the resultant sequence which is far from converging to $1$
Edit : The first question was answered (thanks to Martin R for the link) here
Still looking for hints to my second question . . .
