what is the difference between these two type of convergence and why people saying that that one of them is stronger than the another? what is the difference between these two type of convergence and why people saying that that one of them is stronger than the another?
In another word,how come point-wise convergence is weaker than uniform convergence?
 A: There are sequences of functions which converge pointwise but do not converge uniformly. Pointwise convergence simply means that for every $x$, the sequence of numbers $f_n(x)$ converges. Uniform convergence means that each sequence of numbers $f_n(x)$ converges and that we can choose a convergence rate which does not depend on $x$.
The classic example of a sequence which converges pointwise and not uniformly is $f_n(x)=x^n$ on $[0,1]$. This converges pointwise to a function which is $0$, except at $1$ where it is $1$. But it does not do so uniformly, roughly because if $x$ is very close to $1$ then $n$ must be very large in order for $x^n$ to become small.
We care about the distinction because given uniform convergence, limits of continuous functions remain continuous and many kinds of limits can be interchanged. This does not hold in general given only pointwise convergence.
A: This is because one implies the other; namely, the stronger condition, uniform convergence, implies the weaker condition, point-wise convergence.
If a sequence of functions $\{f_n\}$ converge uniformly to $f$, then for any given $\epsilon > 0$ there exists a single $N$ such that $\sup_x |f_n(x) - f(x)| < \epsilon$ for all $n \geq N$. Clearly this implies that at each $x$, $|f_n(x) - f(x)| < \epsilon$ for every $n \geq N$, so $\{f_n\}$ also converges to $f$ pointwise.
