A slightly different argument using the sequence $x=(1,0,1,0,1,0,\dots)$.$\newcommand{\I}{\mathcal I}\newcommand{\Ilim}{\operatorname{\I-lim}}\newcommand{\Flim}{\operatorname{\mathcal F-lim}}\newcommand{\Glim}{\operatorname{\mathcal G-lim}}$

This sequence is almost convergent to $1/2$. At the same time, it is not difficult to show that if this sequence has $\I$-limit of some ideal $\I$, then the $\I$-limit can only be $0$ or $1$.

 * We can use the fact that $\I$-limit of a sequence is a [cluster point](https://en.wikipedia.org/wiki/Limit_point#For_Sequences_and_Nets) of that sequence. (This holds for any admissible ideal, i.e., for any ideal which contains all finite sets. If we allow also non-admissible ideals, then we can get cluster points or *terms of the sequence* as limits.)
 * For $\I$-convergence we have multiplicativity, i.e., $\Ilim (x_ny_n)=\Ilim x_n\cdot\Ilim y_n$. In particular, for our sequence $x$ we have $x^2=x$. Consequently, if $L$ is an $\I$-limit, then we get $L^2=L$.  

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This is basically just a reformulation of Lorenz's criterion for almost convergent sequence, but since you're looking at connection between almost convergent ideals, I'll mention that a sequence is almost convergent to $L$ if and only if
$$\Flim_n \Glim_k \frac{x_k+\dots+x_{k+n-1}}n=L$$ for any free filters $\mathcal F$, $\mathcal G$.

You can check the section on almost convergent sequences of the paper
<cite authors="Jerison, Meyer">_Jerison, Meyer_, [**The set of all generalized limits of bounded sequences**](http://dx.doi.org/10.4153/CJM-1957-012-x), Can. J. Math. 9, 79-89 (1957). [ZBL0077.31004](https://zbmath.org/?q=an:0077.31004), [MR83697](https://mathscinet.ams.org/mathscinet-getitem?mr=83697).</cite> I have notes on some part of the paper where the results are expressed using ultrafilters rather than using nets available here: [notes](http://thales.doa.fmph.uniba.sk/sleziak/texty/rozne/pozn/clanky/jerisonreferat.pdf), [slides](http://thales.doa.fmph.uniba.sk/sleziak/texty/rozne/pozn/clanky/jerisonslides.pdf).