Prove that lim(x->infinity):x[1/x] = 0 by epsilon-delta defintion. (WITHOUT USING THE SQUEEZE THEOREM)
The Attempt at a Solution
well, It's easy to prove that x[1/x] approaches 0 as x goes to infinity using the squeeze theorem, but the question is to prove that without using the squeeze theorem. I wrote down the epsilon-delta definition and then I tried to use this property that [x]<= n <-> x<= n but I failed to show that for any arbitrary epsilon there exists a N>0 such that x>N -> |x[1/x] - 1| <epsilon.