# sequence - limit superior

• Feb 16th 2010, 10:53 AM
inthequestofproofs
sequence - limit superior
I need to prove the following:
Let Sn be a sequence in R. If s in R and for every e > 0, there exists n' in N s.t. Sn < s + e for all n >= n', prove that Lim superior Sn <= s

Here is how I am approaching it:

first, Sn - s < e -> s is the limit of Sn as n-> infinity.

I also know that Lim Superior Sn = inf (sup[Sn:n>=k])

So, maybe I could use both to say that Lim Sup (Sn) is either s or less than s

What do you think?
• Feb 16th 2010, 01:47 PM
Drexel28
Quote:

Originally Posted by inthequestofproofs
I need to prove the following:
Let Sn be a sequence in R. If s in R and for every e > 0, there exists n' in N s.t. Sn < s + e for all n >= n', prove that Lim superior Sn <= s

Here is how I am approaching it:

first, Sn - s < e -> s is the limit of Sn as n-> infinity.

I also know that Lim Superior Sn = inf (sup[Sn:n>=k])

So, maybe I could use both to say that Lim Sup (Sn) is either s or less than s

What do you think?

Let $S$ be the set of all subsequential limits of $\{S_n\}$ then $\sup\text{ }S=\limsup\text{ }S_n$ try working with that.