What is the cardinality of the set of all sequences in $\mathbb{R}$ converging to a real number?

First note that there are only $2^{\aleph_0}$ sequences of real numbers. This is true because a sequence is a function from $\mathbb N$ to $\mathbb R$ and we have $$\left|\mathbb{R^N}\right|=\left(2^{\aleph_0}\right)^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=2^{\aleph_0}$$

Now note that take any injective sequence which converges to $a$, then it has $2^{\aleph_0}$ subsequences. All are convergent and they all converge to $a$.

Therefore we have at least $2^{\aleph_0}$ sequences converging to $a$, but not more than $2^{\aleph_0}$ sequences over all, so we have exactly $2^{\aleph_0}$ sequences.