[Spivak Calculus, Ch. 5 P. 9] Showing equality of two limits

[WisheDeom]

Homework Statement

Prove that ##\lim_{x \rightarrow a} f(x) = \lim_{h \rightarrow 0} f(a+h)##.

Homework Equations

By definition, if ##\lim_{x \rightarrow a} f(x) = l## then for every ##\epsilon > 0## there exists some ##\delta_1## such that for all x, if ##0<|x-a|<\delta_1## then ##|f(x)-l|<\epsilon##.

Similarly, if ##\lim_{h \rightarrow 0} f(a+h) = m## then for every ##\epsilon > 0## there exists some ##\delta_2## such that for all h, if ##0<|h-0|<\delta_2## then ##|f(a+h)-m|<\epsilon##.

The Attempt at a Solution

I'm really not sure how to go from here. I think maybe I need to perform a proof by contradiction by assuming that ##l \neq m##, but I don't know what kind of contradiction I'm looking for.

 

[voko]

I think you need to prove that if the LHS has some limit l, then the RHS also has that same limit, and vice versa.

 

[E'lir Kramer]

Edit: actually, not quite sure what he's asking for here.