If $f_n$ converges uniformly then $\cos(t)f_n$ converges uniformly?

Question

In my Fourier Series course, it seems the following result is used:

If a series of function $\sum a_n(t)$ converges uniformly then the sequence of functions $\cos(t)\sum a_n(t)$ converges uniformly as well.

I doubt this is true in general. However it is valid when the series $\sum a_n(t)$ converges normally .

Can someone share thoughts about this ?

Answer 1

Note that $$ |\cos(x)f_n(x)-\cos(x)f(x)|\leq|f_n(x)-f(x)|, $$ since $\cos$ is bounded, so if the right hand side goes to 0 uniformly, so does the left hand side.

In general: If $f_n$ converges uniformly to some $f$, and if $g$ is bounded, then $gf_n$ converges uniformly to $gf$.