To whom do you want to prove you can write?
Yourself
Someone in particular
the World at large

If you are embarking down the path of 'I'm going to prove to Myself/My Crazy Ex/My Family/My Friends/My Cat Felix that I can write', it's really important to figure out WHY you want to prove this, and to figure out What Milestone or Achievement would prove this

Is a specific number of books sold the target?
Is it to get offered a book deal?
Is it to get a reader write in and say she/he loves your book?

It's actually pretty easy to prove you can write -
Step 1: Write a good book. Make sure you put effort into making the story smooth. Make sure you use an editor and polish the book
Step 2: Get a good cover. Set a low price as you're currently a new author and need to offer a bargain to attract readers. You can not compete with King and Koontz on Day 1
Step 3: Publish your book in Kindle Store, B&N Nook Store, and Apple Store. Initially go wide as you don't know where your 'perfect best fit' readers are
Step 4: Wait and see if some sales come in. If they do, then you've proven you can write. Someone actually paid money for your book
Step 5: If sales are slow, then consider spending a small amount on marketing. Start with a small $10 to $50 promotion package. Drive some authors to your book page. Get some sales
Step 6: After sales come in, whether by themselves (organically) or via marketing, wait a few weeks for the first reviews. Getting a review is the second part of proving you can write - someone cared enough about your book to review it

There are two rough tests to proving you can write -
Test 1: People are paying their hard earned money for your book
Test 2: People care enough to write a review

At this point, after passing these two 'tests' you can decide whether you want to aim for something more, or if your primary motivation is satisfied and now you want to put your book on auto-pilot and just enjoy the feeling of proven you can write and sell