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.