# Homework 02
## Exercise 1
Let $\left\{ x_n \right\}$ be a real sequence with the limit $a$.
Prove that $\left\{ x_n \right\}$ converges to $a$ **iff**
*every* subsequence of $\left\{ x_n \right\}$ also converges to $a$.

### Proof of $\rightarrow$
Suppose that $\left\{ x_n \right\}$ converges to $a$.
Let $\left\{ x_{n_k} \right\}$ be a subsequence of $\left\{ x_n \right\}$.

Assume that $\varepsilon > 0$.
There exists a positive integer $N$ such that

$\forall n \in \mathbb{N} \ \left( n \ge N
\rightarrow \left| x_n - a \right| < \varepsilon \right)$.

Suppose that $k \ge N$.
Since $\left\{ n_k \right\}$ is a strictly increasing real sequence,
it follows that $n_k \ge k \ge N$. Therefore

$\left| x_{n_k} - a \right| < \varepsilon$.

### Proof of $\leftarrow$
Suppose that every subsequence of $\left\{ x_n \right\}$ converges to $a$.
Since $\left\{ x_n \right\}$ is a subsequence of itself,
we conclude that it converges to $a$.



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