Limit definition and infinitely often

[adnaps1]

Limit definition and "infinitely often"

If we have a sequence of real numbers $x_{n}$ converging to $x$, that means $\forall \epsilon > 0, \exists N$ such that $|x_n - x| < \epsilon, \forall n \geq N.$

So, can we say $P (|x_n - x| < \epsilon \ i.o.) = 1$ because for $n \geq N$, $|x_n - x| < \epsilon$ always holds?

 

[Stephen Tashi]

So, can we say $P (|x_n - x| < \epsilon \ i.o.) = 1$

Is that notation suppose to denote a probability? It doesn't define a probability until you establish a scenario that specifies at least one random variable and its probability distribution. Are you thinking of "picking an x_i at random"? Or did you mean the $x_i$ to be a sequence of real valued random variables instead of a sequence of real numbers? (If the $x_i$ are random variables, you have to use a different definition of limit than the one you gave.)

 

[adnaps1]

Yes, the notation $P(\cdot)$ was supposed to denote a probability, and I want the $x_i$ to be a sequence of real numbers, not random variables. But I understand what you're saying. I cannot talk about probabilities without having a random variable.