A convergent sequence {X(n)} is one for which there exists n0 ∈ ℕ such that for all n≥n0, and a given ε>0, |X(n)-lim X(n)|<ε; and a bounded sequence is one for which there exists M≥0, such that |X(n)|<M for all n ∈ ℕ. Now the boundedness certainly "makes sense" for all n≥n0, but why does the sequence X(n) have to be bounded for any 0<n<n0? Can someone point out whether I am misinterpreting the definition of a sequence of that of convergence or boundedness of a sequence?
[Update]
I was getting confused about the existence of a maximum element out of the X(n) where n<n0, and was wondering whether there was a piecewise defined sequence such that for n=k