September 19, 2006
Theorem 1.
lim sup n →∞
( x n
+ y n
) ≤ lim sup n →∞ x n
+ lim sup n →∞ y n
.
There are many different (equivalent!) definitions of upper limit — also known as limit superior.
Definition 1.
Upper limit of x n a subsequence of x n is the largest number converging to L .
L such that there is
Definition 2.
Upper limit of x n is such a number L that for any ε > 0 the inequality x n
> L + ε is satisfied finitely x n
> L − ε is satisfied infinitely often.
many times and the inequality
Definition 3.
Upper limit of x n is
L = lim k →∞ sup n>k x n
!
.
If we allow limits to be equal to ±∞ then each sequence will have an upper limit (not so for limit). Also, if the limit of x n is equal to the limit.
exists, the upper limit
For our proof the best definition is the last one although proofs based on the other definitions can also be easily constructed. Here is the actual proof.
Proof.
For any k
0
> k , x k
0
≤ sup n>k x n
.
Similarly, for any k
0
> k , y k
0
≤ sup n>k y n
.
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Adding two inequalities together, we get x k
0
+ y k
0
≤ sup n>k x n
+ sup n>k y n
.
Since all x k
0
+ y k
0 are less then the right-hand side, same is true for the maximum of any number of them, or for their supremum.
sup k
0
>k
( x k
0
+ y k
0
) ≤ sup n>k x n
+ sup n>k y n
.
Taking limit of both sides of the inequality (the limits exist!), we get lim k →∞ sup k
0
>k
( x k
0
+ y k
0
)
!
≤ lim k →∞ sup n>k x n
!
+ lim k →∞ sup n>k y n
!
, which produces the desired result, according to the last definition.
2