# Understanding Uniform Convergence of a Sequence of Functions

I am currently self-studying and came across the following theorem:

Suppose that a sequence of functions $$\phi_n$$ converges uniformly to $$0$$ on $$[a,b]$$. Now suppose we have a sequence of functions $$f_n$$ and a function $$f$$ on $$[a,b]$$ such that $$|f_n(x)-f(x)|\le \phi_n(x)$$ for all $$x$$ in $$[a,b]$$. It is stated that $$f_n$$ converges uniformly to $$f$$ on $$[a,b]$$.

However, I'm having difficulty understanding why this is obvious. Could someone please explain this to me? Any help would be greatly appreciated.

• (Uniformly) $f_n→f$ iff $f_n−f→0$; and if $|g_n|≤ϕ_n$ and $ϕ_n→0$ then $g_n→0$. Commented May 4 at 21:00

$$\phi_n$$ converges uniformly to $$0$$ on $$[a, b]$$ means for all $$\epsilon > 0$$ there exists $$N \in \mathbb{N}$$ such that

$$n \geq N \implies \lvert \phi_n(x) \rvert < \epsilon$$

for all $$x \in [a, b]$$. Observe

$$\lvert f_n(x) - f(x) \rvert \leq \phi_n(x) \implies \lvert f_n(x) - f(x) \rvert \leq \lvert \phi_n(x) \rvert$$

Therefore

$$n \geq N \implies \lvert f_n(x) - f(x) \rvert < \epsilon$$

which is the definition of uniform convergence of $$f_n$$ to $$f$$.

For any function $$g:[a,b]\to\Bbb R$$, define $$\|g\|:=\sup\{|g(x)|:x\in[a,b]\}\in[0,\infty].$$ By hypothesis, $$\|f_n-f\|\le \|\phi_n\|$$ and $$\|\phi_n\|$$ converges to $$0$$, hence so does $$\|f_n-f\|$$.