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Homework Statement
Doing some studying for my midterm and came across these problems ...
a)
[itex] f : D \rightarrow R[/itex] with [itex] a \leq f(x) \leq b[/itex] for all c in D\{c}.
Show that if [itex] lim_{x \rightarrow c} f(x)[/itex] exist then [itex] a \leq lim_{x \rightarrow c} f(x) \leq b[/itex]
b) Same thing except we have [itex] g(x) \leq f(x) \leq h(x)[/itex] and [itex] lim_{x \rightarrow c} g(x) = lim_{x \rightarrow c} h(x) = L [/itex]
I need to show [itex] lim_{x \rightarrow c} f(x)= L[/itex].
The Attempt at a Solution
Is this as easy as I think or am I supposed to be more rigorous about the proof ?
a)
[tex] a - L \leq lim_{x \rightarrow c} f(x) -L \leq b - L[/tex]
[tex] a - L \leq 0 \leq b - L[/tex]
Thus,
[tex] a \leq L \leq b [/tex]b) Same "proof" as in part 1.
:(
Will this suffice ?