There exists a real number such that an integer is greater than it, right?

Hello,

I'm confused about something.

-If there's a value "a" such that it's a real number, and there exists a "b" such that it's an integer and that b > a

-If there exists an "a" such that it's a real number and that, for all of "b" that's a real number, b> a

Both of these statements should be true, right? Because no matter what you can always add 1 more to any numbers. But apparently the first statement I listed is true, but the second is false.