Prove triangle inequality for inner product $\langle a,c\rangle \leq \langle a,b\rangle + \langle b,c\rangle $

Question

For all $a,b,c$ which are vectors, determine whether $\langle a,c\rangle \leq \langle a,b\rangle + \langle b,c\rangle $ is true or false.

I have considered Cauchy-Schwartz inequality, but there is a $||b||^2$ left on right hand side, I cannot prove it after that, how can I eliminate $||b||^2$ to make the $\langle a,c\rangle \leq \langle a,b\rangle + \langle b,c\rangle $ is true?