let f,g be continous functions from R to R and suppose that f(x)=g(x) for all rational points. prove that f(x)=g(x) for all x in R.(adsbygoogle = window.adsbygoogle || []).push({});

- i said that we know that since given any real number c, there exists a rational sequence (xn) such that xn converges to c, therefore we conclude that f(xn)=g(xn)=f(c)=g(c), and c is any real number, QED.

is this a good proof?

what about if we only knew that f(1/n)=g(1/n) for all n a natural number?

- i said we know that f(0)=g(0) since the sequence 1/n converges to 0, but other than that, not much else....am i missing something?

prove that for every c a real number, there exists a rational sequence converging to c.

- im not sure how to prove this, my professor said to use the Nested Intervals Property, but I cant really establish the correct intervals

prove that for every c a real number, there exists an irrational sequence converging to c.

- again, not too sure about this one

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# Sequences Help

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