- #1
dylanm
- 3
- 0
Hi folks. I'm a longtime lurker who is starting to explore proof-based mathematics, and I'm having trouble figuring out what I can and cannot do in a problem. I'm stuck on this simple problem from the wonderful Spivak Calculus book:
If [tex]f(x)\le g(x) \forall x[/tex], then [tex]\lim_{x\to a}f(x)\le\lim_{x\to a}g(x)[/tex]
Intuitively, this is obvious. But when I fiddle with it, taking [tex]g(x)-f(x)\ge0[/tex] as my function and proving that the limit of this function, [tex]c=\lim_{x\to a}g(x)-\lim_{x\to a}f(x)[/tex], is [tex]\ge0[/tex], I become stuck. No amount of algebra seems to give me a clean relation between zero and c. I know that one can say that c can be made arbitrarily close to a value of a function that must be [tex]\ge0[/tex], but I'm not sure if I'm allowed to use this sort of thinking in a chapter (5) that has just introduced the concept of a limit.
If [tex]f(x)\le g(x) \forall x[/tex], then [tex]\lim_{x\to a}f(x)\le\lim_{x\to a}g(x)[/tex]
Intuitively, this is obvious. But when I fiddle with it, taking [tex]g(x)-f(x)\ge0[/tex] as my function and proving that the limit of this function, [tex]c=\lim_{x\to a}g(x)-\lim_{x\to a}f(x)[/tex], is [tex]\ge0[/tex], I become stuck. No amount of algebra seems to give me a clean relation between zero and c. I know that one can say that c can be made arbitrarily close to a value of a function that must be [tex]\ge0[/tex], but I'm not sure if I'm allowed to use this sort of thinking in a chapter (5) that has just introduced the concept of a limit.