What inf means in the confidence intervals from a paired sample in t.test() ? R-related question

I'm doing the t.test from a paired sample, only considering as alternative hypothesis that the mean is greater than 0. Why the confidence interval has a minimum value but not a maximum?

The only reason I can think of it is that the maximum value of the mean is not as important as the minimum value if your restriction is Ho>0, but seems strange.

This is the code:

healthy<- c(-0.9914, 1.471, 1.2459, 0.4024, 0.0325, -0.6396, 0.7246, 0.0604)
lame<-c(4.3541, 4.7865, 6.1945, 10.7383, 3.3007, 4.8678, 7.8965, 3.9338)

t.test(lame, healthy, paired = TRUE, alternative='greater')


And this is the outcome:

Paired t-test

data:  lame and healthy
t = 6.5639, df = 7, p-value = 0.0001574
alternative hypothesis: true mean difference is greater than 0
95 percent confidence interval:
3.891736      Inf
sample estimates:
mean difference
5.4708**


Your idea that "the maximum value of the mean is not as important as the minimum value if your restriction is" $$H_\text{alt}>0$$* is pretty much what's going on here. Your asking for a one-sided test has effectively ruled out the idea that the alternative could include $$H_\text{alt}<0$$. Any finding of a value below 0 would be attributed to random variation rather than a true negative value. So there's no need to consider the upper confidence limit for the mean; you just want to make sure that the lower 5% doesn't include the value of 0.
*In standard terminology, $$H_0$$ stands for the null hypothesis, so I use $$H_\text{alt}$$ to represent the alternate hypothesis.