This is the formulation: Let $X_n,n=1,2,...$ be independent, equally distributed random variables. $EX_k=a$(expectation) $k=1,2,...$. For this sequence of $X_n$ the law of large numbers applies: $$\frac{1}{n}\sum_{k=1}^{n}X_k \to^{P}a.$$
Proof: Let $t$ be a real fixed number. (With $f_x(t)$ I denote the characteristic function of variable X). So $$f_{\frac{1}{n}\sum_{k=1}^{n}X_k}(t)=\prod_{k=1}^{n}f_{X_k}(\frac{t}{n})=\text{(this equality is unclear to me }=(1+ai\frac{t}{n}+o(\frac{t}{n}))^n\to^{n\to \infty}\text{this is unclear to me also }=e^{ait}$$
This definitely has something to do with macloren's series of $e^x.$ I just can't see it clearly. I would very much appreciate if someone could do a step by step clarifications of these unclear inequalities to me.
We have $f_{X}(t) = 1 + i t a + o(t)$ as $t \rightarrow 0$.
Characteristic functions, however, have these two general properties: $$ f_{ {1 \over n} X(t)} = f_{X}(t/n) \quad \mbox{and} \quad f_{ X(t) + Y(t) } = f_{X}(t) f_{Y}(t). $$ (i.e., rescaling the original r.v. rescales the argument of the char. fun., and sums of r.v.'s are transformed into products of char. fun.'s).
The first equality that is unclear to you follows from these two properties.
The last conclusion (limit) you are asking about is obtained by dropping the negligible term $o(t)$ and examining the limit $$ \lim_{n \rightarrow +\infty} \; (1 + {1 \over n} i a t)^{n}. $$ And it is one of these: https://en.wikipedia.org/wiki/List_of_limits#Notable_special_limits