
Proof question
Let X be a proper subset of R and let f: X to R. Prove (from the definition) that if f(x) tends to L as x tends to infinity then f(x) tends to L as x tends to infinity.
My attempt:
(For all epsilon > 0) (There exists k > 0) (For all x E X) x>k implies f(x)  L < epsilon
So epsilon< f(x)L < epsilon
Similarly, if f(x) tends to L as x tends to infinity then
(For all epsilon > 0) (There exists k > 0) (For all x E X) x>k implies f(x) + L < epsilon
epsilon < f(x) + L < epsilon which is the same as epsilon< f(x)L < epsilon
So if f(x) tends to L, f(x) tends to L
Is this correct?
