What is the Limit Inferior of the sequence ${x}_{n}$?

Question

$x_n = 1 + (-1)^n + \frac{1}{3^n}$

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Answer to the above question is Two. Please tell me the proof. Thank you!

Answer 1

Let $A$ be the set of accumulation points of $(x_n)$. Then

$ \lim \inf x_n= \min A$.

We have $x_{2n} \to 2$ as $ n \to \infty$ and $x_{2n-1} \to 0$ as $ n \to \infty$ . Hence $A=\{0,2\}$.

Answer 2

Hint:

$$\liminf_{n\to\infty} x_n =\lim_{n\to\infty}\Bigl(\mkern1mu\inf_{k\ge n}x_k\Bigr)=\lim_{n\to\infty}\biggl(\inf_{k\ge n}\Bigl(\mkern1mu\inf_{k\text{ odd}}x_k,\inf_{k\text{ even}}x_k\Bigr)\biggr).$$