As the title says I have to first proof the existence and the calculate the following limit:
$$\lim_{n\to\infty} \int_R \frac{1+nx}{(1+x)^n } 1_{[0,1]} (x) d\lambda$$
My idea:
Because of the indicator-function I can rewrite the integral and go from $R$ to the interval $[0,1]$:
$$\lim_{n\to\infty} \int_0^1 \frac{1+nx}{(1+x)^n } d\lambda$$
because $(1+x)^n > 1+nx $ follows from the Bernoulli inequality
$\frac{1+nx}{(1+x)^n}<1 =:M$ and the existence follows from the dominated convergence Theorem.
Can I the continue this way? :
$$\lim_{n\to\infty} \int_0^1 \frac{1+nx}{(1+x)^n } d\lambda= $$
$$\int_0^1 \lim_{n\to\infty} \frac{1+nx}{(1+x)^n } d\lambda= \int_0^1 0 d\lambda = 0$$