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Calculus/Real Analysis; use sequences show continuos

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Expert: - 11/25/2009


Question
given a sequence {an} n=1 -> oo
a function f: N->R  where f(n) = an

prove that every sequence of real numbers is continuous.

Answer
Are N and R suppose to both be the set of real numbers?
To say that the function is continuous, that means finding the delta such that
if |x1-x2| < delta, |f(x1)-f(x2)| < epsilon.

Let delta, the absolute value of the difference in the x values, be epsilon/(n+1).
It can be seen that |f(x1) - f(x2)| = |n(x1) - n(x2)| = n|x1-x2|.
Since |x1-x2|<epsilon/(n+1), we have |f(x1)-f(x2)| = n(epsilon/(n+1)).
This can clearly be seen to be less than epsilon.

Thus, we have shown that f is a continuos mapping from one set of reals to another set of reals.

Calculus

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