Website content copied - How can I prove that I wrote it?

You can consider notifying about your site updates on social media (Twitter, Facebook etc.) as soon as you post. The timestamp recorded there can be a fair indicator that you wrote first.

Assuming, popular search engines already index your web-pages regularly (use the site operator, site:example.com, to find out) the date in the cached copy can be used as a rough indicator of when the content was published.

Update: Through Google Webmaster Tools, you can enforce a setting to have Googlebot crawl your site too often or manually ask Google to crawl a specific page or site.

As this article suggests, you can write a polite email to your competitors to desist from copying & then if that doesn't help, send a formal DMCA letter to his web hosting company and also possibly to his advertising partner(s).

I've a good idea.

Suppose you create a web page you want to prevent copying. Link this page from nowhere in your site. But create some links from other sites which your competitor would not know so that the search engines would index them.

After some time after say Google has indexed the page you can create internal links to the page in your site. So now if this page is found in competitor's site, search engines would know that it's copied content and would downgrade it with "copy" penalty. You can request removal of such "copied" pages from Google etc.

Another way is to make visible your sitemap page (say sitemap.xml) to only search engines IP address. Googlebot, Bingbot have fixed IP addresses. They'll be able to see your site pages and index them. Don't include those pages in local search until indexed. Better use Google custom search.

If you have a financial interest in protecting your copy, I would recommend consulting a lawyer who is familiar with prosecuting copyright infringement in your jurisdiction.