I got a probability problem I can solve, but my solution does not use an assumption which is given in the formulation of the problem. I am afraid that this is might be a sign that my solution is incorrect. Could you please take a look and let me know whether the assumption is indeed needed.
$X_1, X_2,...$ are i.i.d. random variables, $X_1$ has a continuous pdf $f(x)$ such that $\lambda = \lim_{0 \leftarrow x}f(x)>0$ and $f(x)>0$ $\forall x \in (0, \infty)$, $Z_n=n* \min(X_1,...,X_n)$, show that $Z_n \rightarrow Z$ in distribution and $Z$ is exponentially distributed with parameter $\lambda$. My concern that I can show it without using the continuity of $f$.
We consider $x>0$, because for $x \le 0$ $P(Z_n>x)=1$.
$P(Z_n>x)=P(n* \min(X_1,...,X_n)>x)=P(X_1>\frac{x}{n})^n=(1-F_{X_1}(\frac{x}{n}))^n=(1-\int_0^{\frac{x}{n}}f(t)dt)^n$
I show that $\lim_{n \to \infty}P(Z_n>x)=e^{-\lambda x}$ $\forall x>0$.
I change value of $f$ just at point $0$, I say $f(0)=\lambda$. The changed $f$ is still pdf for $X_1$, and the changed pdf is right continuous, i.e. $\lim_{0 \leftarrow x}f(x)=\lambda=f(0)$.
Then $\forall\epsilon>0$ $\exists \delta$ such that $\forall$ $0<t<\delta$ $|\lambda - f(t)|<\epsilon$ $\Rightarrow$ $\lambda - \epsilon < f(t) < \lambda + \epsilon$. I find $n_0$ such that $\frac{x}{n_0}<\delta$, then $\forall n>n_0$:
$\int_{0}^{x/n}(\lambda - \epsilon)dt < \int_{0}^{x/n}f(t)dt<\int_{0}^{x/n}(\lambda + \epsilon)dt$.
Thus:
$\frac{(\lambda - \epsilon)x}{n}<F(\frac{x}{n})<\frac{(\lambda + \epsilon)x}{n}$,
and
($1-\frac{(\lambda - \epsilon)x}{n})^n>(1-F(\frac{x}{n}))^n>(1-\frac{(\lambda + \epsilon)x}{n})^n$.
It follows that:
$e^{-(\lambda - \epsilon)x} \ge \lim_{n \to \infty}P(Z_n>x) \ge e^{-(\lambda +\epsilon)x}$.
Because $\epsilon$ can be as close to $0$ as we wish, $\lim_{n \to \infty}P(Z_n>x)=e^{-\lambda x}$ and $Z$ is exponentially distributed with parameter $\lambda$.
As you see I nowhere use the continuity of $f$, what I am doing wrong?
Hint:
Assuming Riemann integration, how do you justify the existence of integrals like $$ \int\limits_{0}^{x/n}f(t)\,\mathrm{d}t\,? $$