Lim inf/sup innequality question

[transgalactic]

$$\liminf _{n->\infty} x_n+\limsup _{n->\infty} y_n\leq \limsup _{n->\infty} (x_n+y_n)\leq\limsup _{n->\infty} x_n+\limsup _{n->\infty} y_n\\$$

proving the first part:
$$\limsup _{n->\infty} (x_n+y_n)\leq\limsup _{n->\infty} x_n+\limsup _{n->\infty} y_n\\$$

lim sup is the supremum of all the limits of the subsequences

this is true because of some law regarding the sum of two subsequences

correct??

 

[transgalactic]

why this is true
x_n and y_n are bounded (as n->infinity)
lim sup x_n >=lim x_r_n
lim sup y_n >=lim y_r_n
lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n=lim(x_r_n+y_r_n)
on what basis they get to lim(x_r_n+y_r_n) from sum of limits??

and then
lim(x_r_n+y_r_n)=limsup(x_n +y_n)
how they got to this conclusion??

 

[transgalactic]

whats the theory behind these operations
??

 

[Office_Shredder]

why this is true
x_n and y_n are bounded (as n->infinity)
lim sup x_n >=lim x_r_n
lim sup y_n >=lim y_r_n
lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n=lim(x_r_n+y_r_n)
on what basis they get to lim(x_r_n+y_r_n) from sum of limits??

Because (this has to be one of the first things you learned about limits), the sum of two limits is equal to the limit of the sum of their arguments (or whatever the thing inside the limit is called). It's a pretty straightforward proof that you should try to do if you haven't seen it.

and then
lim(x_r_n+y_r_n)=limsup(x_n +y_n)
how they got to this conclusion??

Considering you haven't told us what x_r_n is supposed to be, it's impossible to answer this

 

[transgalactic]

x_r_n is a subsequence of x_n
y_r_n is a subsequence of y_n

lim(x_r_n+y_r_n)=limsup(x_n +y_n)
how they got to this conclusion??

 

[transgalactic]

how to understand this last part?

 

[transgalactic]

in the solution they say that
because (x_r_n +y_r_n) is convergent then it equals limsup(x_n +y_n)

so what that it is a convergent?

 

[transgalactic]

why its true