This is the contradiction that proves that l = m.

[Miike012]

Prove: If f approaches l near a and f approaches m near a, then l = m.

...Im skipping to the end of the proof...

" to comlete the proof a particular ε>0 has to be choses for which the two conditions
|f(x) - l|< ε and |f(x) - m|< ε cannot both hold if l=/=m."

if l=/=m so that |l - m|> 0 , we can chose ε to be |l - m|/2... How did they decide this |l - m|/2??

It follows that there is a δ>0 such that for all x,

If 0<|x-a|<δ, then |f(x) - l|< |l - m|/2
and |f(x) - m|< |l - m|/2.


|l - m| = | l - f(x) + f(x) - m|≤ |l - f(x)| + | f(x) - m| < |l - m|/2 + |l - m|/2 = |l - m|, a contradiction.

can some one explain to me the contradiction?

 

[Joffan]

You're using the definition of a continuous function, which requires that a sufficiently small change (δ) in x will produce less than a small change (ε) in f(x), and this can be achieved no matter how small ε is.

The contradiction exhibited is |l - m| < |l - m| which is achieved by making ε small enough that even two function evaluations cannot bridge the gap between two limit values for the function.