Help with pointwise convergence

Question

for a function $g_n:[0,5] \rightarrow \mathbb{R}$ where
$g_n=-nx^2+5\; if\; 0\le x \le 1/n$
$g_n=0\; if\; 1/n < x \le 5$
show pointwise convergence

My attempt: I am very new to this subject but I thought for pointwise convergence I take $n\rightarrow \infty$ and I get $g=-\infty x^2+5\; for \; 0 \le x \le 0$, so this function is true for x = 0, but infinity multiply zero is problematic. How to deal with this?

Note that $\infty$ is not a number, so you cannot simply plug it into your equation. — Frank Lu, Feb 12 '15 at 00:48

score 0 · Accepted Answer · answered Feb 12 '15 at 00:50

For each $x\in(0,5]$, there is always an $n\in\mathbb{N}$ s.t. $x>\frac{1}{n}$, this means $g_k(x)=0$ for all $k>n$, thus $\lim_{k\to\infty}g_k(x)=0$ for all $x\in(0,5]$

If $x=0$, note that $g_n(0)=0$ for all $n\in\mathbb{N}$. Thus $g_n$ converges point wise to the zero function.

Help with pointwise convergence

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