Blood tests can prove that the pregnancy hormone is increasing as expected and confirms a healthy pregnancy; ultrasound can demonstrate the physical well-being of a growing baby by showing a healthy heart rate or by ruling out an ectopic (tubal) pregnancy.
It's true that miscarriage is a fact of life as we know it, and usually it's due to some sort of doomed genetic mismatch; but although most miscarriages begin with first trimester bleeding, first trimester bleeding isn't always indicative of a miscarriage. Doctors always approach first trimester bleeding as a possible miscarriage until a cause can be determined. Usually it has a good outcome. So, although it's understandable how first trimester bleeding can cause a lot of anxiety and worry, your doctor can usually find something unrelated to the pregnancy – and treatable – to blame it on.