You are confusing the difference between I have two books and I have 2 books.

Two books: "War and Peace," "Uncle Vanya."

2 books: 2, 2, 2, 2, 2,

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- Oct 15th 2011, 10:29 AMHartlwRe: Vector Subspaces
- Oct 15th 2011, 10:43 AMHartlwRe: Vector Subspaces
A set that has no entries is empty. A set that has zero entries is finite. A set that has a zero number of entries is empty.

- Oct 15th 2011, 12:05 PMFernandoRevillaRe: Vector Subspaces
Imprecisions of usual language. What about the following?:

**Definition**We say that the sequence $\displaystyle (x_n)_{n\geq 0}$ has finitely many non zero terms iff there exists $\displaystyle p\in\mathbb{N}$ such that $\displaystyle x_n=0$ for all $\displaystyle n\geq p$ .

This is just what all mathematicians interpret with the expression*the sequence $\displaystyle (x_n)_{n\geq 0}$ has finitely many non zero terms*.

Question: Has the zero sequence finitely many non zero elements?. You are going to say**yes**. - Oct 15th 2011, 12:42 PMHartlwRe: Vector Subspaces

I have a sequence which has (only) three non-zero terms. Can the sequence be 0? No

I have a sequence which has__no terms__* which aren't zero. The sequence could be 0, but nothing else.

I have a sequence which has__only__no terms which aren't zero. The sequence has to be zero.

So W is a subspace because it contains only 0.

* a zero number of non-zero terms. zero is a finite number. Reference Plato - Oct 15th 2011, 01:01 PMHartlwRe: Vector Subspaces
- Oct 15th 2011, 02:45 PMCaptainBlackRe: Vector Subspaces