If $(y_n)$ is a bounded sequence in an inner product space, and $(x_n)$ is a sequence converging to zero, prove that $(x_n,y_n)\to 0$. Where $(x_n,y_n)$ is the inner product.
Since $(y_n)$ is bounded there exists an $M$ such that $y_n\leq M$ for all $n$. I think the triangle inequality is helpful, I'm just not sure how to use it.
Your idea is about right, but we won't use the triangle inequality. If $M = 0$ it is trivial. Suppose $M > 0$, with $\|y_n\| < M$ for all $n$. Let $\epsilon > 0$. There exists $n_0$ natural such that $n > n_0$ implies $\|x_n\| < \epsilon/M$, because $x_n \to 0$. So if $n > n_0$ we have:$$|\langle x_n,y_n\rangle -0| = |\langle x_n,y_n\rangle| \stackrel{\color{red}{\rm CS}}{\leq} \|x_n\|\|y_n\| < \frac{\epsilon}{M}M = \epsilon,$$ using Cauchy-Schwarz's inequality. So $\langle x_n,y_n \rangle \to 0$.
(Cauchy-Schwarz must be good for something other than proving the triangle inequality, right?)