Show that $f_n(x) = \frac{\sin{(nx+3)}}{\sqrt{n+1}}$ converges pointwise

Question

I need to show that this sequence function converges pointwise with the help of ($\delta-\epsilon$) definition.

I tried to first split this sequence by sandwich theorem then I am stuck at applying pointwise convergence definition that how should I choose $\epsilon$ and $N$.

Answer 1

$|f_n(x)| \le \frac{1}{\sqrt{n+1}}$ for all $x$ and all $n$. Hence $(f_n)$ converges uniformly (and hence pointwise) on $ \mathbb R$ to $0$.

Can you proceed ?