Quotient space of $\mathcal{l}^{\infty}$

Question

I am trying to understand better the concept of quotient space, and my professor gave us the following excersise:

Consider $\mathcal{l}^{\infty}$ the set of all bounded infinite sequences and $Y=\left\{ (a_j)_{j=1}^{\infty}\in \mathcal{l}^{\infty} : \lim_{j\to \infty}(a_j)=a \right\}$ for some $a$, this is subset of convergent sequences. Now try to understand and give a representation of $\mathcal{l}^{\infty} / Y$.

I am lost with this, if I take $(a_j)+Y = (b_j)+Y$ then $(a_j-b_j)$ is convergent. I am thinking for example in $(1,0,1,0,...)$ and $(0,-1,0,-1,...),$ something like this two sequences "complement" each other, but really I don't know what else can I say about this set. Any ideas would help, thanks.

Answer 1

Intuition: a convergent sequence is more or less a "constant sequence", at least for $n$ larger than some large $N$. So you are saying that "adding / subtracting constants" is something that you don't want to distinguish.

So can you identify each bounded sequence $a_n$ (modulo these constant, or almost-constants) with another sequence $a_n'$ where $\limsup a_n'=0$?