$\lim_{n\to\infty} \sup_{t \in \mathbb{R}} |F_n(t)+F_n((-t)-)-1|=0$ for i.i.d. symmetric $(X_n)_{n \ge 0}$ 
Let $(X_n)_{n \ge 1}$ be i.i.d. and symmetric with respect to $0$ and let $F_1,F_2, \dots $ be their respective CDF. Show that:
$\lim_{n\to\infty} \sup_{t \in \mathbb{R}} |F_n(t)+F_n((-t)-)-1|=0$

[$F_n(t)=\sum_{i=1}^n \mathrm{1}_{(-\infty,t]}(X_i)$ is the empirical distribution function.]
What I have tried so far:
With symmetry, we can write
$F_n((-t)-)=1-F_n(t-)$
So, now, our problem is to show that:

$\lim_{n\to\infty} \sup_{t \in \mathbb{R}} |F_n(t)-F_n(t-)|=0$.

Let $Y_n(t)=1_{(-\infty,t]}(X_n)$ and $Z_n(t)=1_{(-\infty,t)}(X_n)$.
Let $F(t-)= \lim_{s \rightarrow t} F(s)$ and $F_n(t-)= \lim_{s \rightarrow t} F_n(s)$.
Then, we have $\mathbb{E} [Y_n(t)]=\mathbb{P} [X_n \le t] =F(t)$ and $\mathbb{E} [Z_n(t)]=\mathbb{P} [X_n < t] =F(t-)$.
So, with LLN, we have:
$F_n(t)=\sum_{i=1}^n Y_i(t) \rightarrow F(t)$
and
$F_n(t-)=\sum_{i=1}^n Z_i(t) \rightarrow F(t-)$.
Combining these, we get
$|F_n(t)-F_n(t-)| \rightarrow |F(t)-F(t-)|=|\mathbb{P}[X_n=t]| = 0$ which shows our problem.
Could it be done like that? I am grateful for any help. Thank you very much.
 A: Here is what I think the original problem really is:  Let $F_n$ be the empirical cdf of $X_1,\ldots,X_n$, and let $G_n$ be the empirical cdf of  $-X_1, \ldots, -X_n$.  At all points of continuity of $F_n$ and $G_n$ we have $G_n(x) = 1-F_n(-x)$ and $F_n(t)-G_n(t) = F_n(t)+F_n(-t)-1$.  What is required is to show that, with probability $1$, $\sup_t|F_n(t)-G_n(t)|\to 0$.  
Write $|F_n(t)-G_n(t)| = |(F_n(t)-F(t)) - (G_n(t) - G(t))|\le |F_n(t)-F(t)| + |G_n(t)-G(t)|$, where $F$ is the theoretical distribution function for the law of $X_n$ and $G$ the theoretical distribution function for the law of $-X_n$. (So $G(t) = 1-F(-t)$ at all continuity points.)  But the distribution of the $X_i$ is symmetric, so $G(t) = F(t)$ for all $t$.
By Glivenko-Cantelli we know $\sup_t |F_n - F|$ converges to $0$ with probability $1$.  And also that $\sup_t|G_n-G|$ does too.  This implies $$\sup_t|F_n-G_n| \le \sup_t |F_n-F| + \sup_t |G_n-G| \to 0$$ as well.
The use of the unexplained notation $(-t)-$ did not help.
