What is the meaning of the following statement? Let $x_n$ be a sequence that converges to a non-zero limit. Prove that all except finitely many terms $x_n$ are non-zero.
I am trying to use contradiction to prove, but I am confused what "all except finitely many terms $x_n$ are non-zero" means. Does it mean finitely zero's or all are zero? If I don't know the meaning, I can't negate. Could someone explain to me?
 A: 
Prove that all except finitely many terms $x_n$ are non-zero.

You can read this as

Prove that all terms $x_n$ are non-zero, except finitely many [terms $x_n$ are zero].

Basically, they want you to prove that only finitely many $x_n$ can be $0$ if $\lim x_n\neq 0$. This will probably be done easiest with a contradiction: assume there are infinitely many, then for each $N$ there is an $n$ with $n>N$ and $x_n=0$. Can you finish the proof?
A: Your sequence has the same amount of terms than natural numbers: For every $n \in \Bbb{N}$ you have a corresponding $x_n$. All except finite means that every $x_n = 0$ except for a finite number of terms. 
How does this can be negated? What is the negation of $\textit{all except finite}$?

There are infinitely many terms $x_n = 0$ 

To be sure this negation if correct, try thinking what would be the negation of the negation, and verify it is exactly: All except finite $x_n$ are $0$.
A: I see that the "all"  confuses you when it's modified by "except". You can replace 

all except finitely many are nonzero

by

only finitely many are zero

without changing the meaning.
The negation of either statement is

infinitely many are zero

A: $limx_n=a\neq 0 (n\rightarrow \infty)\  iff$ for every $\epsilon>0$ there is $n_0\in \mathbb{N}$ such that if $n>n_0$ then $|x_n-a|<\epsilon$
Take $\epsilon=\frac{|a|}{2}$ ($|a|$ is the distance from $a$ to 0).
We know that there is $n_0\in \mathbb{N}$ such that for every $n>n_0\ \ |x_n-a|<\frac{|a|}{2}$. That implies that for every $n>n_0\ \ x_n>0$ (if it's not clear why this is true, try to draw a real line and the points $a, x_n$ and $\frac{|a|}{2}$ neighborhood of the point $a$). That means that if there are some $x_n=0$, they are among $x_1,\dots,x_{n_0}$. The maximum number of zero elements is $n_0$, which is finite.
