(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

a) Let {s[itex]_{n}[/itex]} and {t[itex]_{n}[/itex]} be two sequences converging to s and t. Suppose that s[itex]_{n}[/itex] < (1+[itex]\frac{1}{n}[/itex])t[itex]_{n}[/itex]

Show that s [itex]\leq[/itex]t.

b) Let f, g be continuous functions in the interval [a, b]. If f(x)>g(x) for all x[itex]\in[/itex][a, b], then show that there exists a positive real z>1 such that f(x)[itex]\geq[/itex]zg(x) for all x[itex]\in[/itex][a, b].

2. Relevant equations

3. The attempt at a solution

Ok, so I've already done part a. I'm trying to figure out part b. I think my ideas are on the right track, but I'm looking for some help fleshing them out a bit more.

Argue by contradiction. Suppose there does not exist z such that f(x)[itex]\geq[/itex]zg(x). I'm not sure exactly what this implies in terms of deriving a contradiction; some general guidance would be appreciated it. Also, I'm using the definition that the notion that f(x) is continuous at c is equivalent to the following:

For every sequence {s[itex]_{n}[/itex]} in the domain of f converging to c, one has

lim, n-->[itex]\infty[/itex] f(s[itex]_{n}[/itex])=f(c). I somehow want to construct an analogous argument to that in part a, but I'm not sure how the z is similar to the (1+[itex]\frac{1}){n}) term. Can anyone help with this?

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# Homework Help: Sequences and continuous functions

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