Pigeonhole principle proof in combinatorics Consider the following problem:

A politician gives speeches over $50$ days, each day he gives at least $1$ speech. Over the $50$ days he gives no more than $75$ speeches. Prove that there is a subset of days (consecutive days) in which the politician gives exactly $24$ speeches.
My proof: 
Let $X_i$ be the first $i$ days, and $\delta _i$ be the number of speeches in $X_i$. according to the pigeonhole principle, there are at least two $X_i,X_j$ such that $\delta_i$ $mod$ $24$=$\delta_j$ $mod$ $24$, and now we can take $A$=$X_j$- $X_i$ assuming $j>i$, and that is the days in which there are $24$ speeches. Note that the number of speeches cannot be $48$ in one day.
Is this proof valid?
 A: As David and echo have indicated, it is possible that $\delta_j - \delta_i$ is equal to $24$, but is also possible that the difference is $48$ or $72$.   
Let $a_k$ be the number of speeches given after $k$ days.  Since the politician gives at least one speech a day and at most $75$ speeches in total, the sequence $\{a_1, a_2, \ldots, a_{50}\}$ is a strictly increasing sequence of positive integers with $a_{50} \leq 75$.  Define $b_k = a_k + 24$.  Then $\{b_1, b_2, \ldots, b_{50}\}$ is also a strictly increasing sequence of positive integers with $b_{50} = a_{50} + 24 \leq 75 + 44 = 99$.  The union of these sequences consists of $100$ positive integers that are at most $99$.  Hence, by the Pigeonhole Principle, the intersection of the two sequences is nonempty.  Thus, there exist $i$ and $j$, with $j > i$ such that $a_j = b_i = a_i + 24$.  Hence, exactly $24$ speeches were made from day $i$ to day $j$.  
A: I think your proof is not enough since $\delta_j - \delta_i$ could be 48 or 72. 
Following your notation, let
$$
\lambda_i = \delta_i\bmod 24 
$$
Easy to see that $\lambda_i \in \{0, 1, 2, \cdots, 23\}, \forall 1 \leq i \leq 50$. Then there must exist $1 \leq i < j < k \leq 50$ such that
$$
\lambda_i = \lambda_j = \lambda_k
$$
according to the Pigeonhole Principle. Thus one of $\delta_j - \delta_i$ and $\delta_k - \delta_j$ should be 24 (note that they can't be greater than 24 at the same time)
A: No it is not: $A$ is the number of speeches in some string of consecutive days (not necessarily in one day), and there is no reason why tbis should not be $48$, or even $72$.
Hint for a correct proof: consider the numbers $\delta_i$ that you have defined, and also the numbers $\delta_i+24$.
