Problem on Continuity of function Prove that if f is a continuous function on [-1, 1] such that f (x) < 1 for all x, then there exists
$k \in N$ such that $f (x) < (k - 1)/k$ for all $x \in [-1, 1]$
I am just wondering what is wrong with my reasoning here:
Let a := 1-f(x) >0 since f(x) <1
Using Archimedean Property, we can find a k such that 0< $\frac{1}{k}<$ a.
Then we will have f(x) < 1 - 1/k.
However, I have not used the property of continuity here.

Your $a$ depends on $x$ so also $k$ depends on $x$ and then it isn't unique.
Now let's give a proof: $f$ is continuous on the compact then it's bounded above by say $M$ and attains this maximum. We have $M<1$ so we can find $k$ such that
$$M<(k-1)/k<1$$
