Question

Let fn(x) = cosnx/√ n , x belong to R, is not pointwise convergent. True or false with full explanation

Answer 1

False.

Note that $\left| \frac{\cos nx}{\sqrt{n}} \right|\leqslant \frac{1}{\sqrt{n}}\rightarrow 0,n\rightarrow \infty$

Thus

$\underset{n\rightarrow \infty}{\lim}f_n\left( x \right) =\underset{n\rightarrow \infty}{\lim}\frac{\cos nx}{\sqrt{n}}=0$

which means $f_n$ is pointwise convergent.