If and are sequences, k=1,2,..., what's the relation between sup{|x_k|+|y_k|} and sup{|x_k}+sup{|y_k|}? equality or inequality?

Question

gorica · Answer

$$\sup\{|x _{k}+y_{k}|\} $$ and $$\sup\{|x_k|\}+\sup\{|y_k|\} $$

terenzreignz · Answer

It looks like a triangle inequality, no? :)

terenzreignz · Answer

While I recognise that that's not a proper answer... hang on...

gorica · Answer

sorry, I have to made some correction

gorica · Answer

it's $$\sup(\{|x_k|\}\cup\{|y_k|\})$$ and $$\sup\{|x_k|\}+\sup\{|y_k|\}$$

terenzreignz · Answer

This should be simpler.  You can treat the sequences as sets (clearly)
So, the supremum of the union of $\large |x_k|$  and  $\large |y_k|$  would be either one of their individual suprema, right?

(specifically, it would be whichever of the suprema of   $\large |x_k|$  or  $\large |y_k|$  is bigger)

gorica · Answer

so, it is$$\sup\{|x_k|\cup|y_k|\}\leq \sup\{|x_k|\}+\sup\{|y_k|\}$$ ?

terenzreignz · Answer

Yeah :D

terenzreignz · Answer

BTW this is SO not a a Precalculus question :D

gorica · Answer

It's calculus then? I didn't see that there's Calculus 1 when I posted this question, so I put it here :D Better here than into Linear algebra, right? :D