Proving that something is a subspace of all the infinite sequences

[gtfitzpatrick]

Homework Statement

let V be the vector space consisting of all infinite real sequences. Show that the subset W consisting of all such sequences with only finitely many non-0 entities is a subspace of V

The Attempt at a Solution

Ok so i have to show 1.Closure under Addition,2. Closure under Scalar Multiplcation

let x=(x$$_{n}$$),y=(y$$_{n}$$) $$\in$$ W
x + y $$\in$$ W closed under addition and since W is a set of finite sequences $$\in$$ V as it consists of infinite sequences

$$\lambda$$ is a scalar and $$\lambda$$x $$\in$$ W closed under scalar multiplication and since W is a set of finite sequences $$\in$$ V as it consists of infinite sequences

I think this proves it

...But if $$\lambda$$ is negative this proof doesn't work,is there something i am missing or have i provided the required proof?
Thanks

 

[CompuChip]

You want to check your definitions for V and W.
W is not closed under addition because the sequences in W are finite and those in V are infinite... you actually told us yourself that V consists of finite sequences.
What you need to check is that when you add two sequences with finitely many non-zero entries, you get something with finitely many non-zero entries.

So either look carefully at your proof, or at your definitions, because it looks like the proof doesn't belong to the exercise.

 

[gtfitzpatrick]

whoops really sorry V consists of all infinite real sequences.

if W only consists of finitely many entries and V consists of infinite does it not follow that W has to be a sub space?

 

[CompuChip]

Yes, it follows, but you still have to show it by checking the properties of a subspace.

For example: is W closed? I.e., if you take two infinite sequences with only finitely many non-zero entries, and you add them, do you get again a sequence with only finitely many non-zero entries?

What else do you have to show?

 

[HallsofIvy]

Suppose u has n non-zero entries and v has m non zero entries. What is the maximum possible number of non-zero entries of u+ v?