A proof by contradiction argument regarding uniform convergence (explain a text please)

Question

Here $P:H \to \mathcal{K}$ is a projection operator from a Hilbert space onto a closed convex subset.

I don't follow the hypothesis of the proof by contradiction argument for the uniform convergence (all else is fine). Would someone tell me how exactly the contradiction hypothesis is formed?

I don't really understand why the last quantity is greater than or equal to $\epsilon$ for all $n$. Isn't the point the uniformity of the convergence of $h$ -- for a contradiction argument, isn't $o(h_n)/\lVert h_n\rVert$ supposed to still go to zero but at a rate that depends on the sequence $h_n$?

Answer 1

Let's try to formalize Lemma 4.5 a bit, so the hypothesis for the contradiction argument can be deduced more easily.

The lemma states that

• for all locally compact cones of increments $S_K(x)$ and
• for all null sequences h_n

we have that $o(h)/\|h\|$ goes to zero as well. This is what is meant by uniformity of the limit, i.e. no matter which cone and which null sequence you choose, you will always end up with the desired convergence behaviour of $P_K$.

Now, suppose for the sake of contradiction that the convergence is not uniform in the above sense. Then,

• there exists a locally compact cone of increments $S_K(x)$ and
• there exists a null sequence h_n

such that $o(h)/\|h\|$ doesn't go to zero, i.e. there exists an $\varepsilon > 0$ such that $\|P_K(x + h_n) - x - h_n\| / \|h_n\| \geq \varepsilon$ for all $n \in \mathbb{N}$.