a) $x_n(t) = t^n, n \in \mathbb{N}$
b) $x_n(t) = sin(nt), n \in \mathbb{N}$
c) $x_n(t) = sin(t+n), n \in \mathbb{N}$
d) $x_n(t) = sin(at), a \in \mathbb{R}$
e) $x_n(t) = t^n, n \in [1,2]$
f) $x_n(t) = arctan(at), a \in \mathbb{R}$
g) $x_n(t) = e^{t-a}, a \in \mathbb{R}, a \geq 0$
I'm not quite sure how to approach this. I have written in my notes that every bounded sequence is precompact (Heine Borel Theorem) so I think for b), c), and d), because the sequences are bounded ($-1 \leq sin \leq 1$) that implies that they are precompact. Similarly, arctan is bounded so the seq in f) is precompact. For the sequence in a), $t \in [0,1]$ so $x_n$ would be bounded by 1 and therefore precompact.
Is this correct? It seems too easy. How would I go about showing precompactness for the rest of the sequences (e, g)?