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.

Hint : $\frac {k} {k-1}$ has limit $1$ as $k$ tends to infinity and use $a$ in the question. 
In your reasoning $k$ depends on $x$, but in question $k$ is independent of $x$. 
We need continuity to make sure that $f$ attain its maximum which is strictly less than $1$ and hence we can choose an "uniform" $k$.
