Rejected a null hypothesis: weird conclusion I am confused. There is a question / claim, which says this:

Original Claim: the percentage of Blue M & M's is greater than 5%. The
  hypothesis test results in a P-value of 0.0010, given alpha = 0.05

So everything is fine so far. I have the P-value and compare it with alpha. Obviously, P-Value < Alpha, so I am rejecting the null hypothesis.
The null hypothesis I used is: H0: P > 0.05... 
Everything still fine...
But, after that, I've asked to state a final conclusion and I see this in the answer model:

There is sufficient evidence to support the claim that the percentage
  of blue M&M's is greater than 5%

Here, you can imagine a facepalm image of me, because this is just weird. I have rejected the null hypothesis, saying that the M & M's are not greater than 5%, thus: how can it say there is evidence they ARE greater than 5% while I have rejected it?
Am I doing something wrong here? Do I get the definitons wrong or something? Is rejecting actually saying it is correct? I don't get it...
Please, can someone explain this to me?
Edit: in my slides, nor my teacher has said that I need an equal sign in my null hypothesis, while Google tells me to do that... so am I correct when I assume that my null hypothesis should be H0: P = 0.05 instead? And when I reject this... the conclusion starts making sense... but am I making sense now?
 A: Without seeing the actual preparation of the question and statistical test I can't say for certain, but I think you may have the null and alternative hypotheses mixed up. The symbols and specific numbers used make this one kind of odd. My understanding of the question is this: the original claim is that more than 5% of the M&Ms are blue. The null would then be that the prevalence of blue M&Ms is 5% or less. I might write it like this:
H0: Prevalence(Blue M&Ms) <= 0.05
H1: Prevalence(Blue M&Ms) > 0.05
These hypotheses are assessed at alpha = 0.05 (not to be confused with the possible proportion of blue M&Ms, which is also 0.05). That means that the hypothesis test is investigating how likely it is that the observed prevalence of blue M&Ms in the sample is consistent with the total population of M&Ms containing more than 5% blue ones. A p-value less than alpha (0.05 here) does lead you to reject the null, supporting the alternative hypothesis.
I do not see any proportions in the setup of the problem that suggest 0.5 in any stage of the solution.
A: Let's put one issue aside: There are bilateral tests (equal vs. not equal) and unilateral tests (smaller or larger). The answer to your question applies in both cases, so don't worry about this right now.
I don't see a way of nudging you in the right direction without telling you the answer here. In your edit, you already mention the right approach, so let me just explain why that is. You have fallen victim to a common semantic misunderstanding:
The researcher has a working hypothesis in mind where there is an effect, that he wants to test for. The null hypothesis is such that it postulates the absence of said effect. The null hypothesis can never be proven, only rejected or failed to be rejected.
The null hypothesis generally reflects what you doubt, not what you think will be the case
So when you read a list of hypotheses that a researcher wants to test, those are almost never the null hypotheses. Oftentimes, those will be -on the contrary- the alternative hypotheses ($H_a$ or $H_1$ if you will).
