Proving the limsup=lim of a sequence

[gottfried]

Homework Statement

Show that the if lim b_n = b exists that limsup b_n=b.

The Attempt at a Solution

Let limsup = L and lim = b

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

Therefore L ≤ b+ε and
|b_n| < L ≤ b+ε

I'm trying to get |b_n-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.

 

[HallsofIvy]

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.