 Solution

A binary prefix code assigns distinct finite binary strings to the source symbols, with no assigned string a prefix of another. It is instantaneous because a complete codeword is recognized without needing the next word's bits.

Associate a word of length $\ell$ with its dyadic subinterval of $[0,1)$ of length $2^{-\ell }$. A prefix relation corresponds to interval containment. Prefix-free words therefore give disjoint intervals, whose total length is at most one. This proves necessity of Kraft inequality.

For sufficiency, sort the requested nonnegative integer lengths increasingly and construct a code greedily in a binary tree. At depth ${\ell }_j$, the preceding codewords forbid exactly ${\sum }_{i<j}{2}^{{\ell }_j-{\ell }_i}$ nodes. The assumed inequality, retaining the positive contribution of the present word, bounds this integer by $2^{{\ell }_j}-1$. Thus at least one node is available; choose it. It is not below an earlier codeword, and the nondecreasing lengths prevent it from being a proper ancestor of an earlier word. Continue to obtain a prefix code of all the required lengths. Hence