Multiplying each term of a sequence converging to $0$ by an integer to get a sequence converging to an arbitrarily chosen number Given a sequence $( x_i )_i$ of real numbers converging to $0$ and $t \in \mathbb{R}$, is it possible to find **integers** $z_i$ such that $\lim_{i \to \infty} z_i x_i = t$?

If an infinite numbers of the $x_i$ are $0$, then it's impossible as stated by Paul Sinclair. So we assume there are only finitely many so, are since we're only interested in the behavior at infinity, we may as well assume for convenience that none of them is $0$.

Define $z_i = \left\lfloor \frac{t}{x_i}\right\rfloor \in \mathbb{Z}$. From the definition of the floor function, we have

$$\left| z_i - \frac{t}{x_i} \right| < 1$$

Multiplying by $|x_i|$, we get

$$|z_i x_i - t| < |x_i| \underset{i \to \infty}{\longrightarrow} 0$$

and we conclude that $x_i z_i$ converges to $t$ at infinity. This is just one example of possible sequence $(z_i)_{i \ge 0}$. Any sequence of integers that are _sufficiently close_ to $\frac{t}{x_i}$ will do (for example, you may replace the floor function with the ceiling function).