How to converge to a bounded real sequence by a sequence of bounded sequences in $l^\infty$ using Banach limits? [duplicate]
Solution 1:
This has nothing to do with Banach limits.
First it is easy to reduce the problem to the case where $x\geq0$ (you first decompose $x$ in its real and imaginary parts, then you write these as the difference of their positive and negative parts).
Now, for each $n$, define $$ x_n(k)=\frac{\lfloor n x_k\rfloor}{n}. $$ What this achieves is to divide $[0,1]$ in $n$ slots, and for each entry one chooses the "closest slot".
By construction, $\|x-x_n\|_\infty\leq\frac1n.$
And $x_n$ only takes the values $0,\frac1n,\frac2n,\ldots,1$.