# Taking the lim operation on both sides

1. Mar 16, 2009

### transgalactic

is it legal to take the limit on both sides of an expression

i particular

when i have

sup x_n <=sup y_n

is it ok to do

limsup x_n <=limsup y_n

is the limit of the supremums group of subsequences is the limsup on the sequence
??

2. Mar 16, 2009

### yyat

One of the properties of the limit is that if $$a_n\le b_n$$ for all n, then $$\lim_{n\to\infty}a_n\le\lim_{n\to\infty}b_n$$ provided that these limits exist.

3. Mar 17, 2009

### Office_Shredder

Staff Emeritus
This isn't quite that case yyat.

Counterexample: xn = 1 if n=1, 0 otherwise
yn = 1/2 for all n

supxn = 1
supyn = 1/2

limsup xn = 0
limsup yn = 1/2

4. Mar 17, 2009

### yyat

I don't see how this contradicts anything I wrote.

But you are right, if transgalactic means that sup x_n=sup{x_n,n>=0}, then the statement is wrong by the counterexample you gave (with x_n, y_n switched). I assumed it meant sup{x_n,n>m}<=sup{y_n,n>m} for all m.

Last edited: Mar 17, 2009