# Proving the limsup=lim of a sequence

1. Nov 7, 2012

### gottfried

1. The problem statement, all variables and given/known data
Show that the if lim bn = b exists that limsup bn=b.

3. The attempt at a solution

Let limsup = L and lim = b

We know for all n sufficiently large
|bn-b|<ε
|bn| < b+ε

Therefore L ≤ b+ε and
|bn| < L ≤ b+ε

I'm trying to get |bn-L|<ε or |L-b|<ε both of which I believe imply that b=L.
The problem is I can't get my absolute value signs to be correct.

2. Nov 7, 2012

### HallsofIvy

Staff Emeritus
I'm not sure why you would do that. The "limsup" of a sequence is defined as the supremum of the set of all subsequential limits. If the sequence itself converges, then every subsequence converges to that limit. That is the "set of all subsequential limits" contains ony a single number.