Finding a limit of recursively given sequence

Given a sequence {an}
∞
n=1
:
a1=2
∀n∈N:an+1=4−
3
an
find limn→∞an
. Writing out first few terms i found out an→3
as n→∞
. But i would like it to be in more precise way (the best would be the ϵ
-definition). So I attempted to get a direct formula for an
not depending on any other ai
's but also couldn't find a suitable way to do that. Could you please give me some hints, how to go about this problem in more mathematical way?

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