I was going through the book *Ultrafilters Throughout Mathematics* and I came across the notion of ultralimits, defined below. 

>**Ultralimit.** Let $(X,\tau)$ be a topological space, $(x_i)_{i\in I}$ be a sequence from $X$ and $\mathcal{U}$ is an ultrafilter on $I$. Then, an ***$\mathcal{U}$-ultralimit*** of $(x_i)_{i\in I}$ is defined to be a point $x\in X$ such that for all $U\in \tau$, $x\in U$ implies that $\{i\in I: x_i\in U\}\in \mathcal{U}$. A sequence $(x_i)_{i\in I}$ from $X$ is said to have an ***ultralimit*** if there exists an $\mathcal{U}$-ultralimit for $(x_i)_{i\in I}$.

I was thinking of a stronger version of this notion, defined below.

>**Strong Ultralimit.** Let $(X,\tau)$ be a topological space, $(x_i)_{i\in I}$ be a sequence from $X$ and $\mathcal{U}$ is an ultrafilter on $I$. Then, a ***strong $\mathcal{U}$-ultralimit*** of $(x_i)_{i\in I}$ is defined to be a point $x\in X$ such that for all $U\in \tau$, $x\in U$ iff $\{i\in I: x_i\in U\}\in \mathcal{U}$. A sequence $(x_i)_{i\in I}$ from $X$ is said to have an ***strong ultralimit*** if there exists an strong $\mathcal{U}$-ultralimit for $(x_i)_{i\in I}$.
   
One of the reasons I am interested in this question is the following criteria of compactness via ultralimits.

>**Theorem.** A topological space $(X,\tau)$ is compact iff every sequence $(x_i)_{i\in I}$ from $X$ and has an $\mathcal{U}$-ultralimit for any ultrafilter $\mathcal{U}$ on $I$. 

My questions are:
1. Is it true that a sequence has ultralimits iff it has strong ultralimits? (I suspect a negative answer to this question, but can't find a counterexample.)
2. Assuming the answer to the above question is negative, what kind of compactness arises iff every sequence $(x_i)_{i\in I}$ from $X$ and has a strong $\mathcal{U}$-ultralimit for any ultrafilter $\mathcal{U}$ on $I$? Is it something already known?

Any help is appreciated.