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(adsbygoogle = window.adsbygoogle || []).push({}); Calculusbook:

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.

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# Homework Help: Simple limit proof

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