- #1
the_kid
- 116
- 0
Homework Statement
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].
Homework Equations
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?