[note: also under discussion in s.o.s. math board]
Is it possible to prove using only the definition of liimit?
By definition of limit,
x_n ->a iff
for ALL ε>0, there EXISTS an integer N such that n≥N => |x_n - a|< ε.
To show that a sequence converges to a certain limit, we need to FIND an integer N such that n≥N => |x_n - a|< ε. How can we construct such an N that works?
But at the beginning of the proof, they said that ε = 1/n????? So I am a little confused...why is ε changing for different values of n? This is really werid...shouldn't ε be given to be some FIXED positive number?
When they say "ε = 1/n" at the beginning of the proof, is this the "same" ε as the one that appears at the end when we're trying to prove the limit using definition? or are they totally unrelated?