In an NIP theory, show that if $I$ is an endless indiscernible sequence and $J$ is a Morley sequence in the type $\lim(I)$, then $I+J^*$ is indiscernible.
Here, $J^*$ means the sequence $J$ with the order reversed.
As we are working in a NIP theory, the type $p=\lim(I)$ is a global complete type.
I know I can reduce this problem to prove that for any $i_1,\dots,i_{n+1}\in I$ and $j_1,\dots,j_m\in J^*$ there exists $j'\in J^*$ (whose index in $J^*$ is less than the index of $j_1$) such that $$i_1,\dots,i_n,i_{n+1},j_1,\dots,j_m\equiv i_1,\dots,i_n,j',j_1,\dots,j_m.$$ I also know that I can assume that $J$ is $I$-indiscernible.
I don't exactly know how to attack this problem. I think I am missing something. Some hints or references will be appreciated.
You're on the right track with your reduction, but you'll find it easier to "move" elements of $J^*$ into $I$, rather than the reverse. Also, it's easier to work formula-by-formula than to realize the complete type all at once.
First, let's observe the following.
Claim: For any formula $\varphi$, any $i_1<\dots<i_n$ in $I$, and any $j'<j_1<\dots<j_m$ in $J^*$, if $$\varphi(i_1,\dots,i_n,j',j_1,\dots,j_m)$$ then there exists $i'>i_n$ in $I$ such that $$\varphi(i_1,\dots,i_n,i',j_1,\dots,j_m).$$
Proof: By the definition of Morley sequence, $j'$ realizes $\lim(I)$ over $I$ and $j_1,\dots,j_m$. By the definition of $\lim(I)$, any formula in this type is realized by an element of $I$ (in fact, by a whole end segment). $\square$
Ok, now fix some $i_1<\dots<i_n$ in $I$ and $j'<j_1<\dots<j_m$ in $J^*$ and assume $$\varphi(i_1,\dots,i_n,j_1,\dots,j_m).$$ Applying the claim $m$ times, we obtain $i_{n+1},\dots,i_{n+m}$ with $i_1<\dots<i_n < i_{n+1} < \dots < i_{n+m}$ such that $$\varphi(i_1,\dots,i_n,i_{n+1},\dots,i_{n+m}).$$ By indiscernibility of $I$, it follows that for any $i_1'<\dots,i'_{n+m}$ in $I$, $$\varphi(i_1',\dots,i_{n+m}').$$ Since $\varphi$ was an arbitrary formula, we conclude that for any $i_1'<\dots,i'_{n+m}$ in $I$, $\text{tp}(i_1,\dots,i_n,j_1,\dots,j_m) = \text{tp}(i_1',\dots,i'_{n+m})$. Since $(i_1,\dots,i_n,j_1,\dots,j_m)$ was an arbitrary increasing sequence from $I+J^*$, we conclude that any such sequence has the same type, and hence $I+J^*$ is indiscernible.