From "$\text{lim inf}_{k\rightarrow\infty} \|x(k)\|=0 $" to "$\text{lim}_{k\rightarrow\infty} \|x(k)\|=0 $" (liminf to lim)

Question

Consider the following:

If $\ \ \text{lim inf}_{k\rightarrow\infty} \|x(k)\|=0 \ \ $ and $\ \ \text{lim}_{k\rightarrow\infty} \|x(k)\|^2=0\ \ $ then $\ \ \text{lim}_{k\rightarrow\infty} \|x(k)\|=0 $

My question is

1. If getting rid of "$\text{lim inf}_{k\rightarrow\infty} \|x(k)\|=0 $ ", does it still hold?

2. Since $\|x(k)\|\geq0$, can we say "$\text{lim sup}_{k\rightarrow\infty} \|x(k)\|=0 $ implies $\text{lim}_{k\rightarrow\infty} \|x(k)\|=0 $"

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If consider the following graph, it follows that "$\text{lim inf}_{k\rightarrow\infty} \|x(k)\|=0 $" does not imply $\text{lim}_{k\rightarrow\infty} \|x(k)\|=0 $ obviously.

Note: In my case, the wave should be above $0$.

Answer 1

All of your statements are correct. It is indeed sufficient to have $\|x(k)\|^2 \to 0$ or $\limsup \|x(k)\| = 0$.