I was planning to show the limit of $c_n$ exists, i.e. $c_n$ is convergent, by proving it's bounded. Since $c_n$ is increasing, $\forall n$ $c_n \geq c_1$ (assuming $c_1$ is the start term of the sequence), so $c_n$ is bounded below. However, I am unsure how to show $c_n$ is bounded above.
Once I've shown $c_n$ is bounded, I can conclude it is convergent and therefore has a limit. I am not sure how to show then that this limit is $\leq L$. I was planning to use the following theorem:
"If a sequence $\{a_n\}$ is convergent, then $a_n \leq M$ for large $n$ $\Rightarrow \lim a_n \leq M$."
Since $c_n$ is convergent, I just need to show $c_n \leq L$. I am not sure how to go about this; I think the fact $c_n \leq d_n$ for all $n$ and the fact $d_n \rightarrow L$ can be used to claim $c_n \leq L$ for all $M$.
For some $N$, we have $d_{n}\leq L+1$ for all $n\geq N$, so $c_{n}\leq\max\{d_{1},...,d_{N},L+1\}$ for all $n$, so $(c_{n})$ is a bounded sequence and hence converges, say, the limit is $M$, then by limit comparison rule, we have $M\leq L$.