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Suppose function f: D 

class="katex-mathml"> R and function g: D  R , with x0 as an accumulation point of D. Functions f and g have limits at x0. Show that if f(x)  g(x) for all x  D, then the lim x  x0 f (x)  lim x  x0 g(x).


I understand the limits exists for both functions, and utilized the definitions of a limit to start the proof; after defining the limit of f(x) = L and the limit of g(x) = M. I've let ϵ = (L - M) ÷ 2.


I'm stuck on the algebra of the functions, please explain the steps. 

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