Prove that all non-recursive languages are infinite

First, not every infinite language is undecidable. Some infinite languages are very easy to decide. For example, the language of all words ($Sigma^*$). The converse is true, however: every undecidable language is infinite. In order to show this, we can simply show how to decide every finite language: Consider a finite language $L={w_1,…,w_n}$. In order to decide it, you can construct a TM $M$ that encodes in its states the words $w_1,…,w_n$. This is done as follows: the machine starts in $q_0$, which intuitively corresponds to the empty word $epsilon$. $M$ then reads an input letter. If there is no letter, meaning that the word is $epsilon$, then $M$ either goes to $q_{acc}$ or $q_{rej}$, depending on whether or not $epsilonin L$. Now, if there is an input letter $sigma$ on the tape, $M$ goes to a state that corresponds to $sigma$. Then, again, it reads the tape. If there are no more letters, it either accepts or rejects, depending on whether $sigmain L$. Otherwise, it read the next input $tau$ and moves to the state $sigmatau$, and so on, until the maximal length of a word in $L$ is reached, in which case $M$ rejects immediately. So essentially, $M$ has a state for each word of length at most $k$, where $k$ is the maximal length of a word in $L$. Thus, $M$ can keep track of the word that was read so far, and decides if it needs to be accepted.