# Determining Pointwise Convergence of Sequences of Functions

Recall from the Pointwise Convergence of Sequences of Functions page that a sequence of functions $(f_n(x))_{n=1}^{\infty}$ with common domain $X$ is said to be pointwise convergent if for all $x \in X$ and for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then:

(1)We will now look at some examples of determining whether a sequence of functions is pointwise convergent or divergent.

For example, consider the following sequence of functions with common domain $\mathbb{R}$:

(2)We claim that this sequence converges pointwise to the limit function $f(x) = 0$. To show this, we note that for all $x \in X$ and for all $n \in \mathbb{N}$ that:

(3)Therefore we see that:

(4)Now take the limit as $n$ goes to $\infty$ from both sides:

(5)Since $\lim_{n \to \infty} - \frac{1}{n^2} = 0$ and $\lim_{n \to \infty} \frac{1}{n^2} = 0$ we have by the Squeeze theorem that then $\lim_{n \to \infty} \frac{\sin (2x + 3n)}{n^2} = f(x)$ where $f(x) = 0$, and so the sequence $(f_n(x))_{n=1}^{\infty}$ converges pointwise to $f(x) = 0$.

For another example, consider the following sequence of functions with common domain $[0, 1]$:

(6)Note that since $0 \leq x \leq 1$ that then $0 \leq x^2 \leq 1$ and $0 \leq 2x \leq 2$, and so for $x \in [0, 1]$ we see that

(7)Therefore we have that:

(8)Taking the limit from both sides as $n \to \infty$ and we get that:

(9)Once again by the squeeze theorem we have that $(g_n(x))_{n=1}^{\infty}$ converges pointwise to the limit function $g(x) = 0$.