Find Limsup and Liminf of fn & gn

[aaaa202]

Homework Statement

Let 1_A stand for an indicator function and
Let fn = 1_{n} and gn={1_{1} n odd, 1_{1} n even.
Find lim_n->∞sup{fn} and limsup_n->∞{gn}

Homework Equations

The Attempt at a Solution

The limits are pointwise so I found given x, then
lim_n->∞sup{fn} = 0
limsup_n->∞{gn} = {1 for x=1 or x=2 and 0 elsewhere}
Do you agree? I just wanted to check basically :)

 

[Office_Shredder]

Your definition of g_n seems to suggest it is always equal to 1_{1}, was this intentional?

 

[aaaa202]

oops no 1_{1} for n odd, 1_{2} for n even

 

[Ray Vickson]

oops no 1_{1} for n odd, 1_{2} for n even

What is fn? Your definition means that fn is a function, with fn(x) = 1 if x = n and fn(x) = 0 for x ≠ n. However, that does not seem to be what you really mean. Just tell us in words what are fn and gn---forget about trying to (mis)use indicator functions.

 

[aaaa202]

fn(x) = {1 if x=n, 0 elsewhere} How am I misusing indicator functions?
By definition an indicator 1_A = {1 x\in A, 0 else}

 

[jbunniii]

I think you have the right answer, but it could be stated better. First, you need to include the argument $(x)$ of the functions, and second, you should try to typeset it so it will be clearer what you mean. I assume you meant the following:
$$f_n(x) = 1_{\{n\}}(x) = \begin{cases}1 & \text{ if }x = n \\
0 & \text{ otherwise}\end{cases}$$
$$g_n(x) = \begin{cases}1_{\{1\}}(x) & \text{ if }n\text{ is even} \\
1_{\{2\}}(x) & \text{ if }n\text{ is odd}\end{cases}\text{ (for all }x\text{)}$$
And I interpreted your answers as:
$$\limsup_{n \rightarrow \infty} f_n(x) = 0 \text{ (for all }x\text{)}$$
and
$$\limsup_{n \rightarrow \infty} g_n(x) = \begin{cases}
1 & \text{ if }x = 1\text{ or }x = 2 \\
0 & \text{ otherwise} \end{cases}$$
You can right click on my equations to see how they are typeset. If you show your work, we can check whether your reasoning is right.

 

[Ray Vickson]

fn(x) = {1 if x=n, 0 elsewhere} How am I misusing indicator functions?
By definition an indicator 1_A = {1 x\in A, 0 else}

Yes, I know what an indicator function is; that is why I wrote what I thought you meant for fn, and it turned out to be correct---that is exactly what you meant.

It is often impossible to tell when reading some messages whether or not the OP really knows what he/she is saying; often people write one thing when they mean another. (However, you could have made everything clear by saying that fn and gn are functions, and by specifying their domain.)