According to the definition of pointwise convergence: A sequence $f_m(x)$ of real valued functions defined on D a subset of real numbers is said to be pointwise convergent to$f(d)$ at a point $d\epsilon D$ if for all $\epsilon$>0 there exists a $m_0$ such that $|f_m(d)-f(d)|$<$\epsilon$ for all $m>m_0$
to check pointwise convergence a given interval for a given function , do we need to do all this work above? or we can simply find the $\lim m\to$infinity for the function $f_m(x)$ and if it exists then say that the function is pointwise convergent?