Is uniform convergence and pointwise convergence the same in function space C[X,R]?

Question

In GF Simmon's book on Topology, I came across the definitions of point wise convergence of a sequence of functions and uniform convergence of a sequence of functions.

For all future references, $C[X,R]$ is the set of continuous bounded real functions defined on the metric space $X,$ and $B$ is the subset of all bounded real functions on $X.$

Let $\{f_n\}$ be a sequence of real functions defined on $X.$

Pointwise convergence is when if for each point $x\in X,$ the sequence of real functions $\{f_n(x)\}$ converges to a function $f(x)$, i.e., at each $x,$ for a given $\varepsilon$, there exists a natural $N$ such that for all $n>N, $ $|f_n(x)-f(x)|<\varepsilon$.

Uniform Convergence is when the $\varepsilon$ doesn't depend on $x.$ Which means that a natural $N$ can be found for each $\varepsilon$ such that it works for all $x\in X.$

In terms of convergence of the sequence of functions to a function, Pointwise means that for a given $\varepsilon$ you will find an $N$ for each $x\in X.$ Uniform means that for a given $\varepsilon$ you will find an $N$ which works for all $x\in X.$ Am I right?

Answer 1

Well, strictly speaking, the quantifiers in your last paragraph are not okay. Actually, it should be: "Pointwise mean that for a $\varepsilon>0$, and for each $x\in X$, there exists an $N\in\mathbb N$ such that...". Can you see the difference? We talk first about the point $x\in X$. Your understanding of uniform convergence is correct.

With respect to the title of you question, these kind of convergence are not the same, even when $X$ is compact. Consider $X=[0,1]$, the sequence $f_n(x)=x^n$ converge pointwise to a function which is not even continuous, despite of that, the sequence does not have a uniform limit (since the uniform limit of continuous functions is continuous).