Proving Divergence

[renolovexoxo]

There exists L in Real numbers such that for all E>0 there exists N in Natural numbers such that n>/= N implies |xn-L|< E.

Negate and use this statement to prove the sequence xn=(-1)^n is not convergent.

I have the negation and understand intuitively why it cannot converge, however the proof aspect is giving me some trouble. What is the best way to start?

 

[Freeman]

Negation:∃ e>0 such that ∀ L ε R, n ε N, ∃ m ε N, m>n, |x_M-L|>e.
Proof: Take e=1/2, ∀ L ε R, n ε N
if L<0, let m=2n, then m>n, |x_m-L|=|1-L|>1>1/2
if L>=0, let m=2n+1, then m>n, |x_m-L|=|-1-L|>=1>1/2. QED.

 

[renolovexoxo]

limts of functions

Negate the definition of the limit of a function, and use it to prove that for the function
f : (0; 1) --> R where f(x) 1/x, lim x-->0 f(x) does not exist.

I think I have the negation, but I am having trouble using it.

 

[Freeman]

Defintion: lim(x→a) f(x)=L iff ∀e>0, ∃δ>0: |f(x)-L|<e whenever |x-a|<δ.
Negation: lim(x→a) f(x) does not exists iff ∃e>0: ∀L in R, ∃δ>0, |f(x)-L|>e whenever |x-a|<δ.
Proof: Note that f(x)>1, let e=1, ∀L in R.
If L<0, then ∀δ>0, |f(x)-L|>1 whenever |x|<δ.
If L≧0, let δ=1/(L+1), then |f(x)-L|=|1/x-L|>1 whenever |x|<δ.