Proving limits with epsilon and delta

[LBloom]

Homework Statement

Prove that as x approaches 0, sin(1/x) has no limit.

Homework Equations

|x-a|<d and f(x)-L<e

The Attempt at a Solution

my teacher explained it, but i didnt quite get where the contradiction is at the end. We chose epsilon to be 1/2

 

[n!kofeyn]

Following your teacher's hint, assume that
$$\lim_{x\to 0} \sin(1/x) = L.$$
Then for every ε>0, there exists a η>0 such that 0<|x-0|<η implies |sin(1/x)-L|<ε. Now, let ε=1/2. Then there exists a δ>0 such that 0<|x|<δ implies |sin(1/x)-L|<1/2.

Now that I've started it, can you keep it going until you get a contradiction?