Theorem 1.5.3 of Statistical Inference by Casella and Berger States that the function $F(x)$ is a cdf if and only if the following three conditions hold:
1) $\lim_{x \to - \infty} F(x) = 0 \text{ and } \lim_{x \to \infty} F(x) = 1.$
2) $\ldots$
Now, how would I go to prove the forward direction of this first condition?
By definition we would need to find, for every $\epsilon > 0$, an $N> x \text{ s.t. } |F(x)|< \epsilon$
I am having some problems finding the relationship between $\epsilon$ and $N$ as I would do with a limit of a real function to formally prove a limit. How is this Done?
Thanks in advance
"cdf" = "cumulative distribution function"
"pdf" = "probability density function"
nothing = "cumulative density function". The words "cumulative" and "density" contradict each other. Densities are intensive and cumulative things are extensive.
\begin{align} F(x) = \Pr(X\le x) & = \Pr\left( \left\{ \lfloor x\rfloor \le X < x \right\} \cup \bigcup_{n=-\infty}^{\lfloor x\rfloor-1} \left\{ n \le X<n+1 \right\} \right) \\[10pt] & = \Pr\left( \lfloor x\rfloor \le X < x \right) + \sum_{n=-\infty}^{\lfloor x\rfloor-1} \Pr( n \le X<n+1 ) \le 1. \end{align} So this is a convergent series of non-negative terms. If $\displaystyle\sum_{m=1}^\infty a_m <\infty$ and $\forall m,\ a_m\ge0$, then $$ \lim_{M\to\infty} \sum_{m=M}^\infty a_m = 0. $$ This follows from the fact that $$ \lim_{M\to\infty} \sum_{m=1}^M a_m = \text{the sum of the whole series.} $$