- #1
Krovski
- 11
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Homework Statement
Fix a real number a>1. If r=p/q is a rational number, we define a^r to be a^(p/q). Assume the fact that f(r)=a^r is a continuous increasing function on the domain Q of rational numbers r.
Let s be a real number. Prove that lim r--->s f(r) exists
Homework Equations
For all ε>o there exists δ>0 s.t. |x-x0|<δ implies |f(x)-f(x0)|<ε
Where x0 is an accumulation point for the domain
The Attempt at a Solution
Fix ε>0 and set δ=ε [not sure about this]
With s an accumulation point for Q
|r-s|<δ implies |f(r)-f(s)|<ε
[trouble here]
|f(r)-f(s)|= |a^r - a^s|
[i'm not sure how I should alter this equation to get to |r-s|<δ=ε. Perhaps I am taking the wrong approach? I just need to show that a limit exists as r--->s ]