T/F: If $\limsup x_n = 1$ then $x_n \leq 1$ for sufficiently large $n$

Question

I was asked this question on a homework and I believe the statement is false. My reasoning is, imagine the sequence $x_n = 1 + \frac{1}{n}$ for $n = 1, 2, 3, ...$, which has $\limsup x_n = 1$ but $x_n \neq 1, \ \forall n$. Is this logic okay, or am I missing something?

I meant $1 + \frac{1}{n}$, my mistake

Answer 1

A good counter example could be

$$x_n:=1+\frac{(-1)^n}n\;,\;\;\text{for which}\;\;\limsup_{n\to\infty}x_n=1$$

as this is also the limit of the whole sequence. You could as well take any monotone descending sequence converging to $\;1\;$ , for example $\;\cfrac{n+1}n\;$ , etc.