If $u_{2n}$, $u_{2n+1}$, $u_{3n}$ are convergent sequences then $u_n$ is a convergent sequence The proof is:
Show that if $u_{2n}$, $u_{2n+1}$, $u_{3n}$ are convergent sequences then $u_n$ is a convergent sequence
I don't know where should I start to justify this correctly...
I tried to apply the property that "if $u_{2n}$, $u_{2n+1}$ are convergent than those limits will be the only sublimits of the series", and because of that if $u_{3n}$ is a sublimit then the 3 sublimits are equal and the sequence is convergent... 
But this is not correct I guess because we can have 2 sublimits equal but the other one can be different... 
Can someone give a hint? 
Thanks!

Hint: Write $L$ and $M$ and for the subsequential limits of the even and odd terms. Without more, there is no reason to conclude that $L = M$.
But now notice that $u_{3n}$ alternates between being an even term and an odd term, and is convergent. 
Can you finish from here?
