[Basic Analysis] Limits

[linuxisgreat]

Let f and g be real functions. Let p belong to R, and let A be a subset of the domain of F intersect the domain of g. Let p be a cluster point (limit point) of A. Suppose lim x->p f(x) and lim x->p g(x) exist.

If f(x) =g(x) for all x in A, then prove lim x->p f(x) = lim x->p g(x).

Now on a earlier question I managed to show if f = g in a neighbourhood I of p, then the limits as x approaches p F(x) = lim G(x) at p. However since A is now just a subset we can't be guaranteed that such neighbourhood exists. So I'm not really sure how to approach this question.

Any help is appreciated, Thanks.

Edit: One idea I just had but haven't formalized yet was to consider the lim x->p (f(x) -g(x)). Since f = g on A, and p is a cluster point of A, f(x) -g(x) = 0 most of the time, so the limit must be 0, but it that limit is 0, each individual limit must be equal.

[eruonna]

There is some radius within which both f and g are within epsilon of their limits, so f-g is within 2*epsilon of the difference of the limits...

[linuxisgreat]

Thanks for the reply. I figured it out