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Mathematical Induction over multiple variables
Legionnaired
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Registered User regular
I don't want to post the problem exactly, since I think that'd constitute academic dishonesty...
But let's say that I have something I need to prove for all Integers n >= 3, and for all reals x < -2.
What is the process for proving it for all possible values? Do I do with the reals the same way? Is the base step the same even though it can not, technically, equal -2?
But let's say that I have something I need to prove for all Integers n >= 3, and for all reals x < -2.
What is the process for proving it for all possible values? Do I do with the reals the same way? Is the base step the same even though it can not, technically, equal -2?
Legionnaired on
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May I suggest the old standby of proof by contradiction? Assume that there exist a real value in the domain for which your hypothesis doesn't hold true, and show that some absurd condition follows from that.
It's hard to say in the general case, but you could try focusing on the interaction between x and n. If you have something like k*x*n (where k is some constant or expression not involving x or n) you know x*n will be negative for all applicable values of x and n. If it's j*n - k*x for positive j and k then you know the expression as a whole will be positive, since both J*n and -k*x will be positive.
Does the problem specifically state you should solve the integer part by induction? If not it might be intended for you to simply analyze the behavior of the function at the boundry points (n=3, x=-2) and how it behaves as you increase and/or decrease n and x respectively. If it does, see what you can prove with induction on n and see if you can relate that to the behavior of x.