My instructor's lecture notes state:
Let $\{f_n:E\rightarrow\mathbb{R}\}_{n\in\mathbb{N}}$ be a sequence of functions defined on nonempty $E\subset\mathbb{R}$. We say that the sequence converges pointwisely to $f$ on $E$ if $\forall x\in E,\forall \epsilon>0,\exists N\in \mathbb{N}:\forall n\geq N, |f_n(x)-f(x)|\leq \epsilon $. In this case, $f(x):=\lim_{n\rightarrow \infty}f_n(x)$.
Why was $f$ defined in the last sentence? Doesn't the definition of pointwise convergence in the previous sentences imply that $f(x)=\lim_{n\rightarrow \infty}f_n(x)$ for all $x\in E$ when pointwise convergence holds? I assumed that the notation "$:=$" is used to define $f$.
I agree; “$:=$” means “is defined to be”. It is used when introducing new notation. In this instance, the simple equality sign $=$ seems more appropriate.