Recall that a sequence ( ) converges to L if and only if
Prove that if then .
Suppose , then
Then we know
The limit of the two outside terms is L, and hence the limit of is L by the squeezing theorem for sequences.
Conversely we can say: suppose then
But I doubt this is right or even the right approach. Can anyone show me a better and more rigorous proof?
That was really simple! Much simpler than what I was thinking:
1) Suppose L is positive. Then you can take = L/2 to show that for large enough n, so and the sequence converges to L.
2) Suppose that L is negative. Then you can take = -L/2 to show that for large enough n, so and the sequence converges to -L.
3) Suppose that L= 0. Then [tex]||a_n|- |L||= ||a_n||= |a_n| = |a_n- L|.