# Infs & sups

## Homework Statement

f is a continuous function on [0,1]

M is the set of continuous functions on [0,1] which are 0 at 1 ... i.e. for all m in M, m(1) = 0

I want to know if it's true that

$$|f(1)| = \inf\{\sup\{|f(t)+m(t)|: t \in [0,1]\} : m \in M\}$$

## The Attempt at a Solution

So.... you're choosing t to make it as big as possible and choosing m to make it as small as possible...

If f(1)=0 then you could choose m = -f and then for any t f+m would be zero, so the whole thing would be zero and the equation would be true...

But I'm not sure about the f(1) not equal to 0 case...

$$m(t) = tf(1)-f(t)$$