Let $X=(X_t)_t$ a stochastic process s.t.
1) For all $0\leq t_1<...<t_n<\infty $, $X_{t_{k+1}}-X_{t_k}$ are independent for all $k=1,...,n-1$.
2) For all $0\leq s<t<\infty $, the distribution of $X_t-X_s$ depend on $t-s$ only.
3) $X_0=0$ , $\mathbb E(X_t)=0$ and $\mathbb E(X_t^2)=t$.
4) The distribution of $X$ is scale invariant, that is for all $c>0$ and all $t>0$, $c^{-1}X_{c^2t}$ and $X_t$ has the same distribution.
Prove that $X_{t}-X_s\sim N(0,t-s)$ where $t>s\geq 0$
Attempts
To prove that $X_t-X_s\sim N(0,t-s)$, I set $t_j^{n}=s+\frac{j}{n}(t-s)$, where $j=0,...,n$ and $Y_j^n=X_{t_{j+1}^{(n)}}-X_{t_j^{(n)}}$. So, $$X_t-X_s=\sum_{k=0}^{n-1}Y_{j}^{(n)}=:S_n.$$
I would like to use central limit theorem, but the only thing I have is that $\{Y_j^{(n)}\}_{i=0,...,n-1}$ are i.i.d., not $\{Y_j^{(n)}\}_{\substack{0\leq i\leq n-1\\ n\in\mathbb N}}$. Any idea on how to proceed ?