i am trying to prove that the property x<y implies a^x<a^y for any real numbers x and y. i am having a little trouble using two sequences of rational numbers converging to x and y respectively. using some analysis and the theory of convergent sequences i can show that a^x<=a^y but i cannot show STRICT inequality. could someone help me with this? thanks!
hi, thanks! i am familiar with that definition of exponents and it does make things much easier to deal with.
the definition i am using for a^x is the limit of a^x_n where x_n is a sequence of rational numbers. i am using the fact that every real number is the limit of a sequence of rational numbers.
If are sequences of rational numbers with limits x, y respectively, then for all sufficiently large n it will be true that and . Assuming you can prove that this implies and , it then follows that .