I think that you need to provide more context for the question.
More detail about .
More about what you are proving.
Hi!
I got stuck in an assignment in real analysis.
I have and I would like it to be less than or equal to
Can I do that, and if I can then why? Im assuming it has something to do with triangle inequality but...??
I hated absolute values back in the days and now it comes back biting me in my a##
Sure, the question is
"Let f and g be continuous functions on R. Show that the functions max(f(x), g(x)) and min(f(x), g(x)) are continuous."
So far I have defined h(x)=max(f(x),g(x)) and with the epsilon delta theorem reached the point I described
Now if I get
then IŽd almost be done.
Consider four numbers a, a', b, b'. Assume that |a - b| >= |a' - b'|. Then
||a - b| - |a' - b'|| <= |a - a'| + |b - b'| iff
|a - b| - |a' - b'| <= |a - a'| + |b - b'| iff
|a - b| <= |a - a'| + |b - b'| + |a' - b'| iff
|a - b| <= |a - a'| + |a' - b'| + |b' - b|
Now, |a - b| = |a - a' + a' - b' + b' - b| <= |a - a'| + |a' - b'| + |b' - b|.
If |a - b| < |a' - b'|, then do a similar thing since ||a - b| - |a' - b'|| = ||a' - b'| - |a - b||.