Equivalent definition of $\text{limsup}$

Question

Let $\{x_n\}_{n\in\mathbb{N}}\subseteq\overline{\mathbb{R}}$ a sequence. On some texts the definition of $\text{limsup}$ is as follows:

Definition 1. $$\text{limsup}x_n=\inf_{k\ge1}\bigg(\sup_{n\ge k}x_n\bigg)$$

while other texts other texts give the following definition

Definition 2. $$\text{limsup}x_n=\inf_{k\ge 0}\bigg(\sup_{n\ge k}x_n\bigg)$$

---

Question Are the two definitions equivalent?

Thanks!

Answer 1

As mentioned in the comment, the definition depends on whether your definition of $\Bbb N$ includes zero or not, i.e. if the first index of a sequence is $0$ or $1$. But it makes no difference even if $\Bbb N = \{ 0,1,2,\ldots \}$: For each $k$ is $$ \sup_{n\ge k+1}x_n \le \sup_{n\ge k}x_n $$ because the supremum of a smaller set is taken on the left-hand side. In other words, the sequence $$ \bigg(\sup_{n\ge k}x_n\bigg)_k $$ is decreasing, and it's infimum (which is equal to its limit for $k\to \infty$) does not depend on whether the term for $k=0$ is included or not.